| L(s) = 1 | + 4·4-s − 8·13-s + 12·16-s − 16·17-s + 4·19-s + 8·29-s − 8·31-s + 24·37-s − 8·43-s + 4·49-s − 32·52-s + 16·53-s + 24·59-s + 4·61-s + 32·64-s − 8·67-s − 64·68-s + 16·76-s − 32·79-s − 16·83-s − 8·97-s + 8·101-s − 32·107-s + 12·109-s − 64·113-s + 32·116-s − 32·124-s + ⋯ |
| L(s) = 1 | + 2·4-s − 2.21·13-s + 3·16-s − 3.88·17-s + 0.917·19-s + 1.48·29-s − 1.43·31-s + 3.94·37-s − 1.21·43-s + 4/7·49-s − 4.43·52-s + 2.19·53-s + 3.12·59-s + 0.512·61-s + 4·64-s − 0.977·67-s − 7.76·68-s + 1.83·76-s − 3.60·79-s − 1.75·83-s − 0.812·97-s + 0.796·101-s − 3.09·107-s + 1.14·109-s − 6.02·113-s + 2.97·116-s − 2.87·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.428577602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.428577602\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| good | 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 82 T^{4} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 136 T^{3} + 562 T^{4} + 136 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 20 T^{3} - 146 T^{4} - 20 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1714 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 264 T^{3} + 2162 T^{4} - 264 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 34 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 2520 T^{3} + 17426 T^{4} - 2520 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 8134 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} - 104 T^{3} - 2798 T^{4} - 104 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 3048 T^{3} + 27634 T^{4} - 3048 p T^{5} + 288 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 324 T^{3} - 7042 T^{4} + 324 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 8 T + 32 T^{2} + 88 T^{3} - 2894 T^{4} + 88 p T^{5} + 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 204 T^{2} + 19910 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 16 T + 128 T^{2} + 240 T^{3} - 4174 T^{4} + 240 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 284 T^{2} + 35494 T^{4} - 284 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.29501512797475085966123544662, −7.13588569585663014114336140843, −6.89723847577198610426877296343, −6.82724929963524474710323231495, −6.79334067212668756707761512982, −6.61291326472966388617136616025, −5.96143437341199722836198636853, −5.89126667995914317966160372242, −5.59652841924609334790825675525, −5.54204983701888154934496575567, −5.20548752632221928563298398664, −4.75005338284844830900553054216, −4.52806857079287912577582144176, −4.20807900385427558275043333716, −4.16015494190737281487489252668, −3.97680140801205382686129819548, −3.23304291006816481498824459838, −2.82357140665363153117028514468, −2.78377945845621131275261528742, −2.64382980780510428336483881294, −2.21061334529709178231934295255, −2.04028655996066984937320819319, −1.72625833000394906372956533614, −1.02231951697759098449684668105, −0.48620519971689950062011241633,
0.48620519971689950062011241633, 1.02231951697759098449684668105, 1.72625833000394906372956533614, 2.04028655996066984937320819319, 2.21061334529709178231934295255, 2.64382980780510428336483881294, 2.78377945845621131275261528742, 2.82357140665363153117028514468, 3.23304291006816481498824459838, 3.97680140801205382686129819548, 4.16015494190737281487489252668, 4.20807900385427558275043333716, 4.52806857079287912577582144176, 4.75005338284844830900553054216, 5.20548752632221928563298398664, 5.54204983701888154934496575567, 5.59652841924609334790825675525, 5.89126667995914317966160372242, 5.96143437341199722836198636853, 6.61291326472966388617136616025, 6.79334067212668756707761512982, 6.82724929963524474710323231495, 6.89723847577198610426877296343, 7.13588569585663014114336140843, 7.29501512797475085966123544662
