| L(s) = 1 | + 0.574·2-s + 1.63·3-s − 1.67·4-s − 1.06·5-s + 0.938·6-s − 4.08·7-s − 2.10·8-s − 0.327·9-s − 0.612·10-s − 6.08·11-s − 2.73·12-s − 4.68·13-s − 2.34·14-s − 1.74·15-s + 2.13·16-s + 3.83·17-s − 0.188·18-s − 1.93·19-s + 1.78·20-s − 6.67·21-s − 3.49·22-s + 4.65·23-s − 3.44·24-s − 3.86·25-s − 2.68·26-s − 5.43·27-s + 6.82·28-s + ⋯ |
| L(s) = 1 | + 0.406·2-s + 0.943·3-s − 0.835·4-s − 0.477·5-s + 0.383·6-s − 1.54·7-s − 0.745·8-s − 0.109·9-s − 0.193·10-s − 1.83·11-s − 0.788·12-s − 1.29·13-s − 0.626·14-s − 0.450·15-s + 0.532·16-s + 0.930·17-s − 0.0443·18-s − 0.444·19-s + 0.398·20-s − 1.45·21-s − 0.744·22-s + 0.970·23-s − 0.703·24-s − 0.772·25-s − 0.527·26-s − 1.04·27-s + 1.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04133347359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04133347359\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 - T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 - 0.574T + 2T^{2} \) |
| 3 | \( 1 - 1.63T + 3T^{2} \) |
| 5 | \( 1 + 1.06T + 5T^{2} \) |
| 7 | \( 1 + 4.08T + 7T^{2} \) |
| 11 | \( 1 + 6.08T + 11T^{2} \) |
| 13 | \( 1 + 4.68T + 13T^{2} \) |
| 17 | \( 1 - 3.83T + 17T^{2} \) |
| 19 | \( 1 + 1.93T + 19T^{2} \) |
| 23 | \( 1 - 4.65T + 23T^{2} \) |
| 29 | \( 1 + 8.35T + 29T^{2} \) |
| 31 | \( 1 + 3.00T + 31T^{2} \) |
| 37 | \( 1 + 0.680T + 37T^{2} \) |
| 41 | \( 1 + 3.92T + 41T^{2} \) |
| 43 | \( 1 + 11.8T + 43T^{2} \) |
| 47 | \( 1 - 4.03T + 47T^{2} \) |
| 53 | \( 1 + 4.40T + 53T^{2} \) |
| 59 | \( 1 + 2.35T + 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 - 5.15T + 67T^{2} \) |
| 73 | \( 1 - 7.95T + 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 2.41T + 83T^{2} \) |
| 89 | \( 1 + 2.81T + 89T^{2} \) |
| 97 | \( 1 + 7.15T + 97T^{2} \) |
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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
−7.79535352802227616948141538786, −7.43658505001475836297349621791, −6.42789401692395199370744541760, −5.36719749280657235801434971554, −5.25187339702038052031984384707, −4.07995842558172643350119664691, −3.26174751020152630497341483657, −3.08841018178261731042519898520, −2.18426787666141304253203505745, −0.082018169962167583005913665667,
0.082018169962167583005913665667, 2.18426787666141304253203505745, 3.08841018178261731042519898520, 3.26174751020152630497341483657, 4.07995842558172643350119664691, 5.25187339702038052031984384707, 5.36719749280657235801434971554, 6.42789401692395199370744541760, 7.43658505001475836297349621791, 7.79535352802227616948141538786
