| L(s) = 1 | + 1.67e7·2-s + 1.12e11·3-s + 2.81e14·4-s − 1.14e16·5-s + 1.88e18·6-s − 1.91e20·7-s + 4.72e21·8-s − 2.26e23·9-s − 1.92e23·10-s − 4.46e25·11-s + 3.16e25·12-s − 3.31e26·13-s − 3.21e27·14-s − 1.29e27·15-s + 7.92e28·16-s + 6.28e29·17-s − 3.80e30·18-s − 2.42e31·19-s − 3.23e30·20-s − 2.15e31·21-s − 7.48e32·22-s − 3.19e31·23-s + 5.31e32·24-s − 1.76e34·25-s − 5.55e33·26-s − 5.24e34·27-s − 5.39e34·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.230·3-s + 0.5·4-s − 0.0861·5-s + 0.162·6-s − 0.377·7-s + 0.353·8-s − 0.947·9-s − 0.0609·10-s − 1.36·11-s + 0.115·12-s − 0.169·13-s − 0.267·14-s − 0.0198·15-s + 0.250·16-s + 0.448·17-s − 0.669·18-s − 1.13·19-s − 0.0430·20-s − 0.0870·21-s − 0.965·22-s − 0.0138·23-s + 0.0813·24-s − 0.992·25-s − 0.119·26-s − 0.448·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(50-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+49/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(25)\) |
\(\approx\) |
\(2.256985424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.256985424\) |
| \(L(\frac{51}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.67e7T \) |
| 7 | \( 1 + 1.91e20T \) |
| good | 3 | \( 1 - 1.12e11T + 2.39e23T^{2} \) |
| 5 | \( 1 + 1.14e16T + 1.77e34T^{2} \) |
| 11 | \( 1 + 4.46e25T + 1.06e51T^{2} \) |
| 13 | \( 1 + 3.31e26T + 3.83e54T^{2} \) |
| 17 | \( 1 - 6.28e29T + 1.95e60T^{2} \) |
| 19 | \( 1 + 2.42e31T + 4.55e62T^{2} \) |
| 23 | \( 1 + 3.19e31T + 5.30e66T^{2} \) |
| 29 | \( 1 - 1.04e36T + 4.54e71T^{2} \) |
| 31 | \( 1 + 3.13e36T + 1.19e73T^{2} \) |
| 37 | \( 1 - 2.90e38T + 6.94e76T^{2} \) |
| 41 | \( 1 - 1.06e39T + 1.06e79T^{2} \) |
| 43 | \( 1 + 1.60e39T + 1.09e80T^{2} \) |
| 47 | \( 1 - 1.73e41T + 8.56e81T^{2} \) |
| 53 | \( 1 + 1.15e42T + 3.08e84T^{2} \) |
| 59 | \( 1 - 4.05e43T + 5.91e86T^{2} \) |
| 61 | \( 1 - 2.95e43T + 3.02e87T^{2} \) |
| 67 | \( 1 + 3.58e44T + 3.00e89T^{2} \) |
| 71 | \( 1 - 3.76e45T + 5.14e90T^{2} \) |
| 73 | \( 1 + 4.95e45T + 2.00e91T^{2} \) |
| 79 | \( 1 + 2.86e46T + 9.63e92T^{2} \) |
| 83 | \( 1 - 9.05e43T + 1.08e94T^{2} \) |
| 89 | \( 1 - 4.88e47T + 3.31e95T^{2} \) |
| 97 | \( 1 + 6.39e48T + 2.24e97T^{2} \) |
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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
−11.10114639405717469536119137752, −10.05473545282185921639366882329, −8.532716899833121379222795697671, −7.55443546051189370778813808472, −6.15802421313422485071030302175, −5.30640708422525316040841808930, −4.06103302684063759850543689968, −2.88019931043785116701525379255, −2.23998003477643538591394338025, −0.51503788565814098790388542119,
0.51503788565814098790388542119, 2.23998003477643538591394338025, 2.88019931043785116701525379255, 4.06103302684063759850543689968, 5.30640708422525316040841808930, 6.15802421313422485071030302175, 7.55443546051189370778813808472, 8.532716899833121379222795697671, 10.05473545282185921639366882329, 11.10114639405717469536119137752
