| L(s) = 1 | − 4·2-s + (9.44 + 12.4i)3-s + 16·4-s + 32.9i·5-s + (−37.7 − 49.6i)6-s + 193.·7-s − 64·8-s + (−64.6 + 234. i)9-s − 131. i·10-s − 33.0·11-s + (151. + 198. i)12-s − 210. i·13-s − 773.·14-s + (−409. + 311. i)15-s + 256·16-s + 1.12e3i·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (0.605 + 0.795i)3-s + 0.5·4-s + 0.590i·5-s + (−0.428 − 0.562i)6-s + 1.49·7-s − 0.353·8-s + (−0.266 + 0.963i)9-s − 0.417i·10-s − 0.0824·11-s + (0.302 + 0.397i)12-s − 0.346i·13-s − 1.05·14-s + (−0.469 + 0.357i)15-s + 0.250·16-s + 0.941i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.914 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.628125242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.628125242\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 + (-9.44 - 12.4i)T \) |
| 59 | \( 1 + (-2.34e4 + 1.28e4i)T \) |
| good | 5 | \( 1 - 32.9iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 193.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 33.0T + 1.61e5T^{2} \) |
| 13 | \( 1 + 210. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.12e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.67e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.56e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.32e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 5.85e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 3.79e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 4.35e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.69e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 3.28e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.82e4iT - 4.18e8T^{2} \) |
| 61 | \( 1 + 9.80e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 3.40e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.24e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 2.20e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.68e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.13e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.82e5iT - 8.58e9T^{2} \) |
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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
−10.69010461254650163888266759266, −10.34946278100724343367275553262, −9.087515669716335893955281649705, −8.214045097671578782135812469025, −7.78572445310350907447933247694, −6.34634591440974545327371154809, −5.03587885854271449546084115190, −3.96377500880656880496149711319, −2.61048501743026843619742893389, −1.58720416401804565985290047729,
0.46104244835350934405430851662, 1.59159984321821738891763882797, 2.39786370542895646584063654242, 4.15118595590136723468613033796, 5.40425419427632990807528981011, 6.74333554348018441089423554243, 7.66286920762184089582334055767, 8.431064977651504788954522220051, 8.923467372621657502332121556386, 10.10157494082139597570175420823
