| L(s) = 1 | + (−8.38e6 − 1.45e7i)2-s + (4.19e11 − 7.26e11i)3-s + (−1.40e14 + 2.43e14i)4-s + (−1.05e17 − 1.81e17i)5-s − 1.40e19·6-s + (2.32e20 − 4.50e20i)7-s + (4.72e21 − 5.24e5i)8-s + (−2.32e23 − 4.02e23i)9-s + (−1.76e24 + 3.05e24i)10-s + (1.98e24 − 3.43e24i)11-s + (1.18e26 + 2.04e26i)12-s − 3.62e27·13-s + (−8.49e27 + 3.93e26i)14-s − 1.76e29·15-s + (−3.96e28 − 6.86e28i)16-s + (−1.26e30 + 2.19e30i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.857 − 1.48i)3-s + (−0.249 + 0.433i)4-s + (−0.788 − 1.36i)5-s − 1.21·6-s + (0.459 − 0.888i)7-s + 0.353·8-s + (−0.971 − 1.68i)9-s + (−0.557 + 0.965i)10-s + (0.0607 − 0.105i)11-s + (0.428 + 0.742i)12-s − 1.85·13-s + (−0.706 + 0.0327i)14-s − 2.70·15-s + (−0.125 − 0.216i)16-s + (−0.906 + 1.57i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(50-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+49/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(25)\) |
\(\approx\) |
\(0.2471306164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2471306164\) |
| \(L(\frac{51}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (8.38e6 + 1.45e7i)T \) |
| 7 | \( 1 + (-2.32e20 + 4.50e20i)T \) |
| good | 3 | \( 1 + (-4.19e11 + 7.26e11i)T + (-1.19e23 - 2.07e23i)T^{2} \) |
| 5 | \( 1 + (1.05e17 + 1.81e17i)T + (-8.88e33 + 1.53e34i)T^{2} \) |
| 11 | \( 1 + (-1.98e24 + 3.43e24i)T + (-5.33e50 - 9.24e50i)T^{2} \) |
| 13 | \( 1 + 3.62e27T + 3.83e54T^{2} \) |
| 17 | \( 1 + (1.26e30 - 2.19e30i)T + (-9.79e59 - 1.69e60i)T^{2} \) |
| 19 | \( 1 + (9.11e30 + 1.57e31i)T + (-2.27e62 + 3.94e62i)T^{2} \) |
| 23 | \( 1 + (-1.12e33 - 1.94e33i)T + (-2.65e66 + 4.59e66i)T^{2} \) |
| 29 | \( 1 - 4.73e35T + 4.54e71T^{2} \) |
| 31 | \( 1 + (-1.58e36 + 2.75e36i)T + (-5.96e72 - 1.03e73i)T^{2} \) |
| 37 | \( 1 + (7.74e37 + 1.34e38i)T + (-3.47e76 + 6.01e76i)T^{2} \) |
| 41 | \( 1 - 1.61e39T + 1.06e79T^{2} \) |
| 43 | \( 1 - 5.84e39T + 1.09e80T^{2} \) |
| 47 | \( 1 + (4.18e40 + 7.24e40i)T + (-4.28e81 + 7.41e81i)T^{2} \) |
| 53 | \( 1 + (1.06e42 - 1.83e42i)T + (-1.54e84 - 2.67e84i)T^{2} \) |
| 59 | \( 1 + (-1.53e43 + 2.66e43i)T + (-2.95e86 - 5.12e86i)T^{2} \) |
| 61 | \( 1 + (3.88e43 + 6.73e43i)T + (-1.51e87 + 2.62e87i)T^{2} \) |
| 67 | \( 1 + (-3.27e44 + 5.66e44i)T + (-1.50e89 - 2.60e89i)T^{2} \) |
| 71 | \( 1 - 2.48e45T + 5.14e90T^{2} \) |
| 73 | \( 1 + (1.80e45 - 3.12e45i)T + (-1.00e91 - 1.73e91i)T^{2} \) |
| 79 | \( 1 + (-2.35e46 - 4.07e46i)T + (-4.81e92 + 8.34e92i)T^{2} \) |
| 83 | \( 1 + 1.08e47T + 1.08e94T^{2} \) |
| 89 | \( 1 + (-6.44e46 - 1.11e47i)T + (-1.65e95 + 2.86e95i)T^{2} \) |
| 97 | \( 1 + 2.95e48T + 2.24e97T^{2} \) |
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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
−9.424856414868245930632353225839, −8.360604137353267880497835317338, −7.83156260225441744357176129247, −6.87890219528840055727539162550, −4.80277149458927838849601369542, −3.86075406078218985267723665665, −2.41851830298935658058141938213, −1.58371165204588633382120019955, −0.71809990949583776874875801302, −0.05800820848534458510767984734,
2.50524713666203510964595445645, 2.85605859425928866774036824881, 4.37515317813724494395380572609, 5.02744288206304917636066857668, 6.78691716371822502587891027185, 7.81394237839967549529416202946, 8.855622420597372015715488533040, 9.830943773919650742936020701545, 10.71942427994115228992326677561, 11.88092640763848032837983117476
