| L(s) = 1 | + 8.79e12·2-s − 8.60e20·3-s + 7.73e25·4-s − 5.73e28·5-s − 7.57e33·6-s + 1.36e36·7-s + 6.80e38·8-s + 4.17e41·9-s − 5.04e41·10-s − 2.71e45·11-s − 6.65e46·12-s + 2.87e47·13-s + 1.20e49·14-s + 4.93e49·15-s + 5.98e51·16-s + 3.10e53·17-s + 3.67e54·18-s − 1.86e55·19-s − 4.43e54·20-s − 1.17e57·21-s − 2.38e58·22-s + 2.58e59·23-s − 5.85e59·24-s − 6.45e60·25-s + 2.53e60·26-s − 8.12e61·27-s + 1.05e62·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.51·3-s + 0.5·4-s − 0.0225·5-s − 1.07·6-s + 0.237·7-s + 0.353·8-s + 1.29·9-s − 0.0159·10-s − 1.35·11-s − 0.756·12-s + 0.100·13-s + 0.167·14-s + 0.0341·15-s + 0.250·16-s + 0.927·17-s + 0.913·18-s − 0.441·19-s − 0.0112·20-s − 0.358·21-s − 0.959·22-s + 1.50·23-s − 0.535·24-s − 0.999·25-s + 0.0711·26-s − 0.441·27-s + 0.118·28-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
| \(L(44)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{89}{2})\) |
|
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 | \( 1 + 8.60e20T + 3.23e41T^{2} \) |
| 5 | \( 1 + 5.73e28T + 6.46e60T^{2} \) |
| 7 | \( 1 - 1.36e36T + 3.33e73T^{2} \) |
| 11 | \( 1 + 2.71e45T + 3.99e90T^{2} \) |
| 13 | \( 1 - 2.87e47T + 8.18e96T^{2} \) |
| 17 | \( 1 - 3.10e53T + 1.11e107T^{2} \) |
| 19 | \( 1 + 1.86e55T + 1.78e111T^{2} \) |
| 23 | \( 1 - 2.58e59T + 2.95e118T^{2} \) |
| 29 | \( 1 - 2.68e63T + 1.69e127T^{2} \) |
| 31 | \( 1 - 6.88e64T + 5.60e129T^{2} \) |
| 37 | \( 1 - 2.93e68T + 2.71e136T^{2} \) |
| 41 | \( 1 + 2.76e70T + 2.05e140T^{2} \) |
| 43 | \( 1 + 2.01e70T + 1.29e142T^{2} \) |
| 47 | \( 1 - 5.35e72T + 2.96e145T^{2} \) |
| 53 | \( 1 + 1.25e75T + 1.02e150T^{2} \) |
| 59 | \( 1 + 4.72e76T + 1.15e154T^{2} \) |
| 61 | \( 1 + 2.91e77T + 2.10e155T^{2} \) |
| 67 | \( 1 - 1.58e78T + 7.38e158T^{2} \) |
| 71 | \( 1 + 4.85e80T + 1.14e161T^{2} \) |
| 73 | \( 1 + 8.17e80T + 1.28e162T^{2} \) |
| 79 | \( 1 - 1.04e82T + 1.24e165T^{2} \) |
| 83 | \( 1 - 4.94e83T + 9.11e166T^{2} \) |
| 89 | \( 1 + 1.60e84T + 3.95e169T^{2} \) |
| 97 | \( 1 + 3.52e86T + 7.06e172T^{2} \) |
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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
−12.24060569269182683283447707053, −11.17418732768670391111284502323, −10.20433697774009455164584443916, −7.83181493691858884938318321540, −6.42321584058934397331075747569, −5.39265483838556770531654622632, −4.61990747071597836415756065717, −2.88947706489248499216351658144, −1.23345665677276392471285777637, 0,
1.23345665677276392471285777637, 2.88947706489248499216351658144, 4.61990747071597836415756065717, 5.39265483838556770531654622632, 6.42321584058934397331075747569, 7.83181493691858884938318321540, 10.20433697774009455164584443916, 11.17418732768670391111284502323, 12.24060569269182683283447707053
