| L(s) = 1 | + (4 + 6.92i)2-s + (19.9 + 34.6i)3-s + (−31.9 + 55.4i)4-s − 323.·5-s + (−159. + 277. i)6-s + (−284. + 492. i)7-s − 511.·8-s + (294. − 509. i)9-s + (−1.29e3 − 2.24e3i)10-s + (119. + 206. i)11-s − 2.55e3·12-s + (−5.81e3 + 5.37e3i)13-s − 4.54e3·14-s + (−6.47e3 − 1.12e4i)15-s + (−2.04e3 − 3.54e3i)16-s + (−1.02e4 + 1.77e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (0.427 + 0.740i)3-s + (−0.249 + 0.433i)4-s − 1.15·5-s + (−0.302 + 0.523i)6-s + (−0.313 + 0.542i)7-s − 0.353·8-s + (0.134 − 0.232i)9-s + (−0.409 − 0.709i)10-s + (0.0269 + 0.0467i)11-s − 0.427·12-s + (−0.734 + 0.678i)13-s − 0.443·14-s + (−0.495 − 0.857i)15-s + (−0.125 − 0.216i)16-s + (−0.505 + 0.874i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0326i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0209038 + 1.27935i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0209038 + 1.27935i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 - 6.92i)T \) |
| 13 | \( 1 + (5.81e3 - 5.37e3i)T \) |
| good | 3 | \( 1 + (-19.9 - 34.6i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + 323.T + 7.81e4T^{2} \) |
| 7 | \( 1 + (284. - 492. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-119. - 206. i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 17 | \( 1 + (1.02e4 - 1.77e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (-4.82e3 + 8.35e3i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-3.91e4 - 6.77e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-6.94e4 - 1.20e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 1.60e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-7.64e4 - 1.32e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (9.27e4 + 1.60e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (4.25e4 - 7.36e4i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + 1.20e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.65e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-1.24e6 + 2.15e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-1.52e6 + 2.63e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.93e5 - 3.35e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.84e6 - 3.18e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 - 1.57e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.29e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.93e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (4.07e6 + 7.05e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (-6.69e5 + 1.16e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.92341939711065749443798217382, −15.41967767633748820432243804053, −14.46619070072303532516761427245, −12.72628481876810223705055795669, −11.54536634732657975384229296971, −9.601381088606370128908176747864, −8.375326073932773628086061679582, −6.81168357888632453050945433734, −4.68103046049712956632384351648, −3.39338269765668062466063672318,
0.56010130648833225088231556862, 2.78085853788222646183678354315, 4.52559608014640163369828050410, 7.03402526272987570584172732105, 8.220528630867470532598130958500, 10.17468374745028682753468871026, 11.61615606252643888825590028128, 12.72046656072473499417664010058, 13.72712299015936029513323788485, 15.05520489389839023901463150020
