| L(s) = 1 | − 8.79e12·2-s + 6.79e20·3-s + 7.73e25·4-s − 4.34e30·5-s − 5.97e33·6-s − 6.14e36·7-s − 6.80e38·8-s + 1.38e41·9-s + 3.82e43·10-s − 1.67e44·11-s + 5.25e46·12-s − 5.67e48·13-s + 5.40e49·14-s − 2.95e51·15-s + 5.98e51·16-s − 3.24e53·17-s − 1.22e54·18-s − 3.75e55·19-s − 3.36e56·20-s − 4.17e57·21-s + 1.47e57·22-s − 8.77e56·23-s − 4.62e59·24-s + 1.24e61·25-s + 4.99e61·26-s − 1.25e62·27-s − 4.75e62·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.19·3-s + 0.5·4-s − 1.70·5-s − 0.845·6-s − 1.06·7-s − 0.353·8-s + 0.429·9-s + 1.20·10-s − 0.0839·11-s + 0.597·12-s − 1.98·13-s + 0.751·14-s − 2.04·15-s + 0.250·16-s − 0.971·17-s − 0.303·18-s − 0.888·19-s − 0.854·20-s − 1.27·21-s + 0.0593·22-s − 0.00510·23-s − 0.422·24-s + 1.92·25-s + 1.40·26-s − 0.682·27-s − 0.531·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(88-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+87/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(44)\) |
\(\approx\) |
\(0.1888957449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1888957449\) |
| \(L(\frac{89}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8.79e12T \) |
| good | 3 | \( 1 - 6.79e20T + 3.23e41T^{2} \) |
| 5 | \( 1 + 4.34e30T + 6.46e60T^{2} \) |
| 7 | \( 1 + 6.14e36T + 3.33e73T^{2} \) |
| 11 | \( 1 + 1.67e44T + 3.99e90T^{2} \) |
| 13 | \( 1 + 5.67e48T + 8.18e96T^{2} \) |
| 17 | \( 1 + 3.24e53T + 1.11e107T^{2} \) |
| 19 | \( 1 + 3.75e55T + 1.78e111T^{2} \) |
| 23 | \( 1 + 8.77e56T + 2.95e118T^{2} \) |
| 29 | \( 1 + 3.59e63T + 1.69e127T^{2} \) |
| 31 | \( 1 - 8.67e64T + 5.60e129T^{2} \) |
| 37 | \( 1 - 1.92e68T + 2.71e136T^{2} \) |
| 41 | \( 1 + 2.04e70T + 2.05e140T^{2} \) |
| 43 | \( 1 + 7.22e70T + 1.29e142T^{2} \) |
| 47 | \( 1 - 4.35e72T + 2.96e145T^{2} \) |
| 53 | \( 1 + 1.03e75T + 1.02e150T^{2} \) |
| 59 | \( 1 - 1.10e77T + 1.15e154T^{2} \) |
| 61 | \( 1 - 4.36e77T + 2.10e155T^{2} \) |
| 67 | \( 1 + 4.97e79T + 7.38e158T^{2} \) |
| 71 | \( 1 - 2.94e80T + 1.14e161T^{2} \) |
| 73 | \( 1 + 7.53e80T + 1.28e162T^{2} \) |
| 79 | \( 1 + 2.58e82T + 1.24e165T^{2} \) |
| 83 | \( 1 + 6.62e82T + 9.11e166T^{2} \) |
| 89 | \( 1 + 8.08e84T + 3.95e169T^{2} \) |
| 97 | \( 1 - 7.41e85T + 7.06e172T^{2} \) |
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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
−12.85736972547466814115812768091, −11.61826515061108004709102453028, −9.930913566401728756371514858450, −8.758427985937132401030280154027, −7.78221580197980130400781843000, −6.86730410984493347909207082338, −4.37236280775379550751523776054, −3.17863575213613941618806631807, −2.34421222146130258906937531787, −0.20386348093953006742400555354,
0.20386348093953006742400555354, 2.34421222146130258906937531787, 3.17863575213613941618806631807, 4.37236280775379550751523776054, 6.86730410984493347909207082338, 7.78221580197980130400781843000, 8.758427985937132401030280154027, 9.930913566401728756371514858450, 11.61826515061108004709102453028, 12.85736972547466814115812768091
