| L(s) = 1 | + 4.29e9·2-s − 5.99e13·3-s + 1.84e19·4-s + 6.16e22·5-s − 2.57e23·6-s − 2.55e27·7-s + 7.92e28·8-s − 1.02e31·9-s + 2.64e32·10-s + 8.24e33·11-s − 1.10e33·12-s + 2.22e36·13-s − 1.09e37·14-s − 3.69e36·15-s + 3.40e38·16-s + 8.10e39·17-s − 4.42e40·18-s − 2.71e41·19-s + 1.13e42·20-s + 1.53e41·21-s + 3.54e43·22-s − 1.66e44·23-s − 4.75e42·24-s + 1.08e45·25-s + 9.57e45·26-s + 1.23e45·27-s − 4.72e46·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0186·3-s + 0.5·4-s + 1.18·5-s − 0.0132·6-s − 0.875·7-s + 0.353·8-s − 0.999·9-s + 0.836·10-s + 1.17·11-s − 0.00934·12-s + 1.39·13-s − 0.619·14-s − 0.0221·15-s + 0.250·16-s + 0.830·17-s − 0.706·18-s − 0.747·19-s + 0.591·20-s + 0.0163·21-s + 0.832·22-s − 0.921·23-s − 0.00660·24-s + 0.401·25-s + 0.987·26-s + 0.0373·27-s − 0.437·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(66-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+65/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(33)\) |
\(\approx\) |
\(4.126165678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.126165678\) |
| \(L(\frac{67}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4.29e9T \) |
| good | 3 | \( 1 + 5.99e13T + 1.03e31T^{2} \) |
| 5 | \( 1 - 6.16e22T + 2.71e45T^{2} \) |
| 7 | \( 1 + 2.55e27T + 8.53e54T^{2} \) |
| 11 | \( 1 - 8.24e33T + 4.90e67T^{2} \) |
| 13 | \( 1 - 2.22e36T + 2.54e72T^{2} \) |
| 17 | \( 1 - 8.10e39T + 9.53e79T^{2} \) |
| 19 | \( 1 + 2.71e41T + 1.31e83T^{2} \) |
| 23 | \( 1 + 1.66e44T + 3.25e88T^{2} \) |
| 29 | \( 1 - 5.45e47T + 1.13e95T^{2} \) |
| 31 | \( 1 - 1.86e48T + 8.67e96T^{2} \) |
| 37 | \( 1 - 2.14e50T + 8.57e101T^{2} \) |
| 41 | \( 1 + 7.22e51T + 6.77e104T^{2} \) |
| 43 | \( 1 - 1.76e53T + 1.49e106T^{2} \) |
| 47 | \( 1 - 2.08e54T + 4.85e108T^{2} \) |
| 53 | \( 1 - 1.83e56T + 1.19e112T^{2} \) |
| 59 | \( 1 - 3.10e57T + 1.27e115T^{2} \) |
| 61 | \( 1 - 1.43e58T + 1.11e116T^{2} \) |
| 67 | \( 1 + 3.92e59T + 4.95e118T^{2} \) |
| 71 | \( 1 + 1.24e60T + 2.14e120T^{2} \) |
| 73 | \( 1 + 2.06e60T + 1.30e121T^{2} \) |
| 79 | \( 1 + 2.95e61T + 2.21e123T^{2} \) |
| 83 | \( 1 - 2.09e62T + 5.49e124T^{2} \) |
| 89 | \( 1 - 1.03e63T + 5.13e126T^{2} \) |
| 97 | \( 1 + 1.75e64T + 1.38e129T^{2} \) |
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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
−14.34593305846313796684655076004, −13.47419099131972485932032975140, −11.93530931973395509040108334330, −10.24708823574178167643630422131, −8.766633873035407594682105347598, −6.35506745690576753653094904462, −5.86303191000229961086496801721, −3.86007208291752441219529835717, −2.56587004394753263773104492944, −1.08431744292981397182448263762,
1.08431744292981397182448263762, 2.56587004394753263773104492944, 3.86007208291752441219529835717, 5.86303191000229961086496801721, 6.35506745690576753653094904462, 8.766633873035407594682105347598, 10.24708823574178167643630422131, 11.93530931973395509040108334330, 13.47419099131972485932032975140, 14.34593305846313796684655076004
