| L(s) = 1 | + 1.64e3·4-s + 2.83e4·9-s + 7.06e4·11-s + 1.50e6·16-s + 1.85e6·19-s + 2.00e7·29-s + 4.93e6·31-s + 4.65e7·36-s − 3.82e7·41-s + 1.16e8·44-s − 1.15e7·49-s + 1.41e7·59-s + 8.86e7·61-s + 9.51e8·64-s + 4.12e8·71-s + 3.04e9·76-s − 9.37e8·79-s − 2.65e5·81-s − 1.27e9·89-s + 2.00e9·99-s + 3.12e8·101-s − 2.92e9·109-s + 3.28e10·116-s − 2.04e9·121-s + 8.11e9·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 3.21·4-s + 1.43·9-s + 1.45·11-s + 5.75·16-s + 3.25·19-s + 5.25·29-s + 0.959·31-s + 4.62·36-s − 2.11·41-s + 4.67·44-s − 2/7·49-s + 0.151·59-s + 0.819·61-s + 7.08·64-s + 1.92·71-s + 10.4·76-s − 2.70·79-s − 0.000685·81-s − 2.14·89-s + 2.09·99-s + 0.298·101-s − 1.98·109-s + 16.8·116-s − 0.866·121-s + 3.08·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(62.49828731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(62.49828731\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + p^{8} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 411 p^{2} T^{2} + 18641 p^{6} T^{4} - 411 p^{20} T^{6} + p^{36} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 28328 T^{2} + 89193454 p^{2} T^{4} - 28328 p^{18} T^{6} + p^{36} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 35316 T + 2892681078 T^{2} - 35316 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 3786168544 T^{2} + \)\(21\!\cdots\!42\)\( T^{4} + 3786168544 p^{18} T^{6} + p^{36} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 332444725052 T^{2} + \)\(52\!\cdots\!94\)\( T^{4} - 332444725052 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 925426 T + 858791487510 T^{2} - 925426 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6341900496476 T^{2} + \)\(16\!\cdots\!54\)\( T^{4} - 6341900496476 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 10003584 T + 52302031706070 T^{2} - 10003584 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 2467260 T + 49371575832542 T^{2} - 2467260 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 23744441004556 T^{2} + \)\(22\!\cdots\!42\)\( T^{4} - 23744441004556 p^{18} T^{6} + p^{36} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 19103448 T + 602984827739166 T^{2} + 19103448 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1577937637342748 T^{2} + \)\(11\!\cdots\!74\)\( T^{4} - 1577937637342748 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 686469728593004 T^{2} + \)\(12\!\cdots\!82\)\( T^{4} - 686469728593004 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 3201641364617996 T^{2} + \)\(11\!\cdots\!74\)\( T^{4} - 3201641364617996 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 7069218 T + 16866494212382134 T^{2} - 7069218 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 44316386 T + 21654336818123658 T^{2} - 44316386 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 60507234625606124 T^{2} + \)\(18\!\cdots\!94\)\( T^{4} - 60507234625606124 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 206493816 T + 58491352612128526 T^{2} - 206493816 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 107795011437047164 T^{2} + \)\(94\!\cdots\!10\)\( T^{4} - 107795011437047164 p^{18} T^{6} + p^{36} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 5930824 p T + 239633073722978334 T^{2} + 5930824 p^{10} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 469718746005377480 T^{2} + \)\(10\!\cdots\!18\)\( T^{4} - 469718746005377480 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 636267396 T + 801539802340191990 T^{2} + 636267396 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1695363093446047420 T^{2} + \)\(18\!\cdots\!78\)\( T^{4} - 1695363093446047420 p^{18} T^{6} + p^{36} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.38982603996987898087019909461, −7.34451281095043262856227203194, −7.09002674633295675996589258171, −6.49698295620486006254348336651, −6.49508902422253670394629381018, −6.47123789748028507547771984811, −6.40262153381981618587910718179, −5.72506861886367600400707774283, −5.31953288064248131773170060835, −5.08211394832851412534305662436, −4.93542519554634253809509005103, −4.36546344319349812303085246847, −4.12230034030685643024698771781, −3.63165174888306594405087308238, −3.47208333792263523525321428621, −2.88234892504873105204645194663, −2.87679974804778238910966833722, −2.53626646369578629719977469627, −2.52998894182471958958777385417, −1.59363714393697117762269165515, −1.45383746661001787042852979093, −1.30696814624160873210565618841, −1.30219106832111753346067448057, −0.77981701218094448146174055694, −0.52673663577914812462444511882,
0.52673663577914812462444511882, 0.77981701218094448146174055694, 1.30219106832111753346067448057, 1.30696814624160873210565618841, 1.45383746661001787042852979093, 1.59363714393697117762269165515, 2.52998894182471958958777385417, 2.53626646369578629719977469627, 2.87679974804778238910966833722, 2.88234892504873105204645194663, 3.47208333792263523525321428621, 3.63165174888306594405087308238, 4.12230034030685643024698771781, 4.36546344319349812303085246847, 4.93542519554634253809509005103, 5.08211394832851412534305662436, 5.31953288064248131773170060835, 5.72506861886367600400707774283, 6.40262153381981618587910718179, 6.47123789748028507547771984811, 6.49508902422253670394629381018, 6.49698295620486006254348336651, 7.09002674633295675996589258171, 7.34451281095043262856227203194, 7.38982603996987898087019909461
