| L(s) = 1 | − 36·7-s − 52·13-s − 84·19-s + 80·25-s + 604·31-s + 184·37-s − 880·43-s − 96·49-s + 656·61-s − 3.05e3·67-s − 312·73-s + 720·79-s + 1.87e3·91-s − 344·97-s − 3.39e3·103-s + 3.38e3·109-s − 4.57e3·121-s + 127-s + 131-s + 3.02e3·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.94·7-s − 1.10·13-s − 1.01·19-s + 0.639·25-s + 3.49·31-s + 0.817·37-s − 3.12·43-s − 0.279·49-s + 1.37·61-s − 5.56·67-s − 0.500·73-s + 1.02·79-s + 2.15·91-s − 0.360·97-s − 3.24·103-s + 2.97·109-s − 3.43·121-s + 0.000698·127-s + 0.000666·131-s + 1.97·133-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + p T )^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 16 p T^{2} + 24462 T^{4} - 16 p^{7} T^{6} + p^{12} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 + 18 T + 534 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 4572 T^{2} + 8634710 T^{4} + 4572 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 16772 T^{2} + 118361542 T^{4} + 16772 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 42 T + 2742 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 12 T^{2} - 393658 T^{4} + 12 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 + 22356 T^{2} - 27485674 T^{4} + 22356 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 302 T + 2650 p T^{2} - 302 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 92 T + 43774 T^{2} - 92 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 227840 T^{2} + 22474730590 T^{4} + 227840 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 440 T + 203686 T^{2} + 440 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 270652 T^{2} + 39420081222 T^{4} + 270652 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2 \wr C_2$ | \( 1 + 438372 T^{2} + 89997219542 T^{4} + 438372 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 598556 T^{2} + 161876495254 T^{4} + 598556 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 328 T + 368086 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 1526 T + 1116358 T^{2} + 1526 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 500316 T^{2} + 133803927398 T^{4} + 500316 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 156 T - 293274 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 360 T + 287790 T^{2} - 360 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 2272508 T^{2} + 1944939568822 T^{4} + 2272508 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 913024 T^{2} + 920290760478 T^{4} + 913024 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 172 T + 1202710 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.65191691000371093910586536975, −6.31497623321680000856506263897, −6.29145154370225907360820580309, −6.14966468825330265178899740107, −5.90341227698964756225300179073, −5.63180767158203896429166681547, −5.18687656309643816127039088363, −5.10490771161957135264713629681, −4.94165267239835760101628120544, −4.69043161417104098283509897995, −4.32548048063333864939777835537, −4.26924507493639001365615677424, −4.25234761431885473704693335374, −3.65179659972239518231400971340, −3.32708996039277804782914562409, −3.23365945430607864741765641157, −3.19220775814200004406469152421, −2.77219022289939646737603286087, −2.50387435272809842687019584100, −2.47310831380870536983156691987, −2.17587970201688029506931360778, −1.62794123582672902760398543752, −1.34826903354317386712701667760, −1.06692389902457062649981811217, −0.971236304285921129016295712155, 0, 0, 0, 0,
0.971236304285921129016295712155, 1.06692389902457062649981811217, 1.34826903354317386712701667760, 1.62794123582672902760398543752, 2.17587970201688029506931360778, 2.47310831380870536983156691987, 2.50387435272809842687019584100, 2.77219022289939646737603286087, 3.19220775814200004406469152421, 3.23365945430607864741765641157, 3.32708996039277804782914562409, 3.65179659972239518231400971340, 4.25234761431885473704693335374, 4.26924507493639001365615677424, 4.32548048063333864939777835537, 4.69043161417104098283509897995, 4.94165267239835760101628120544, 5.10490771161957135264713629681, 5.18687656309643816127039088363, 5.63180767158203896429166681547, 5.90341227698964756225300179073, 6.14966468825330265178899740107, 6.29145154370225907360820580309, 6.31497623321680000856506263897, 6.65191691000371093910586536975
