| L(s) = 1 | + 1.68e13·2-s − 3.69e20·3-s + 1.27e26·4-s + 4.92e30·5-s − 6.20e33·6-s + 5.74e36·7-s − 4.52e38·8-s − 1.87e41·9-s + 8.27e43·10-s + 2.18e45·11-s − 4.71e46·12-s + 1.19e48·13-s + 9.65e49·14-s − 1.81e51·15-s − 2.73e52·16-s + 2.15e53·17-s − 3.14e54·18-s − 3.79e55·19-s + 6.29e56·20-s − 2.11e57·21-s + 3.67e58·22-s − 8.32e58·23-s + 1.67e59·24-s + 1.77e61·25-s + 2.00e61·26-s + 1.88e62·27-s + 7.33e62·28-s + ⋯ |
| L(s) = 1 | + 1.35·2-s − 0.649·3-s + 0.825·4-s + 1.93·5-s − 0.877·6-s + 0.993·7-s − 0.235·8-s − 0.578·9-s + 2.61·10-s + 1.09·11-s − 0.536·12-s + 0.416·13-s + 1.34·14-s − 1.25·15-s − 1.14·16-s + 0.643·17-s − 0.781·18-s − 0.898·19-s + 1.59·20-s − 0.645·21-s + 1.48·22-s − 0.484·23-s + 0.152·24-s + 2.74·25-s + 0.562·26-s + 1.02·27-s + 0.820·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(88-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+87/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(44)\) |
\(\approx\) |
\(5.419274867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.419274867\) |
| \(L(\frac{89}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 1.68e13T + 1.54e26T^{2} \) |
| 3 | \( 1 + 3.69e20T + 3.23e41T^{2} \) |
| 5 | \( 1 - 4.92e30T + 6.46e60T^{2} \) |
| 7 | \( 1 - 5.74e36T + 3.33e73T^{2} \) |
| 11 | \( 1 - 2.18e45T + 3.99e90T^{2} \) |
| 13 | \( 1 - 1.19e48T + 8.18e96T^{2} \) |
| 17 | \( 1 - 2.15e53T + 1.11e107T^{2} \) |
| 19 | \( 1 + 3.79e55T + 1.78e111T^{2} \) |
| 23 | \( 1 + 8.32e58T + 2.95e118T^{2} \) |
| 29 | \( 1 + 2.95e62T + 1.69e127T^{2} \) |
| 31 | \( 1 - 4.80e64T + 5.60e129T^{2} \) |
| 37 | \( 1 - 3.87e67T + 2.71e136T^{2} \) |
| 41 | \( 1 - 2.28e70T + 2.05e140T^{2} \) |
| 43 | \( 1 - 1.58e71T + 1.29e142T^{2} \) |
| 47 | \( 1 - 7.69e72T + 2.96e145T^{2} \) |
| 53 | \( 1 + 3.99e74T + 1.02e150T^{2} \) |
| 59 | \( 1 + 4.47e76T + 1.15e154T^{2} \) |
| 61 | \( 1 + 8.97e77T + 2.10e155T^{2} \) |
| 67 | \( 1 + 1.98e79T + 7.38e158T^{2} \) |
| 71 | \( 1 - 1.05e80T + 1.14e161T^{2} \) |
| 73 | \( 1 + 4.40e80T + 1.28e162T^{2} \) |
| 79 | \( 1 - 1.07e82T + 1.24e165T^{2} \) |
| 83 | \( 1 + 1.67e83T + 9.11e166T^{2} \) |
| 89 | \( 1 + 8.33e84T + 3.95e169T^{2} \) |
| 97 | \( 1 - 1.28e85T + 7.06e172T^{2} \) |
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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
−14.48584215368207333392877988569, −13.91408795308068450251160587638, −12.36237860004498538204423667707, −10.91901841645193090182976095865, −9.071222625546540846761881347188, −6.22337797782015399444550726073, −5.72285169580206900427119708621, −4.47486283822637940380563910286, −2.55258974218499163699045597480, −1.27101621026077824899442848284,
1.27101621026077824899442848284, 2.55258974218499163699045597480, 4.47486283822637940380563910286, 5.72285169580206900427119708621, 6.22337797782015399444550726073, 9.071222625546540846761881347188, 10.91901841645193090182976095865, 12.36237860004498538204423667707, 13.91408795308068450251160587638, 14.48584215368207333392877988569
