| L(s) = 1 | + 1.96·2-s − 3.07·3-s + 1.87·4-s − 1.53·5-s − 6.05·6-s − 0.246·8-s + 6.45·9-s − 3.02·10-s + 5.05·11-s − 5.76·12-s + 4.52·13-s + 4.73·15-s − 4.23·16-s − 5.34·17-s + 12.6·18-s + 1.52·19-s − 2.88·20-s + 9.94·22-s − 8.43·23-s + 0.758·24-s − 2.63·25-s + 8.90·26-s − 10.6·27-s + 8.90·29-s + 9.31·30-s − 6.95·31-s − 7.84·32-s + ⋯ |
| L(s) = 1 | + 1.39·2-s − 1.77·3-s + 0.937·4-s − 0.688·5-s − 2.47·6-s − 0.0871·8-s + 2.15·9-s − 0.957·10-s + 1.52·11-s − 1.66·12-s + 1.25·13-s + 1.22·15-s − 1.05·16-s − 1.29·17-s + 2.99·18-s + 0.348·19-s − 0.645·20-s + 2.11·22-s − 1.75·23-s + 0.154·24-s − 0.526·25-s + 1.74·26-s − 2.04·27-s + 1.65·29-s + 1.70·30-s − 1.24·31-s − 1.38·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.718333291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.718333291\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 1.96T + 2T^{2} \) |
| 3 | \( 1 + 3.07T + 3T^{2} \) |
| 5 | \( 1 + 1.53T + 5T^{2} \) |
| 11 | \( 1 - 5.05T + 11T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 + 5.34T + 17T^{2} \) |
| 19 | \( 1 - 1.52T + 19T^{2} \) |
| 23 | \( 1 + 8.43T + 23T^{2} \) |
| 29 | \( 1 - 8.90T + 29T^{2} \) |
| 31 | \( 1 + 6.95T + 31T^{2} \) |
| 37 | \( 1 + 0.805T + 37T^{2} \) |
| 41 | \( 1 - 6.05T + 41T^{2} \) |
| 43 | \( 1 + 4.04T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 0.547T + 53T^{2} \) |
| 59 | \( 1 + 3.37T + 59T^{2} \) |
| 61 | \( 1 - 7.20T + 61T^{2} \) |
| 67 | \( 1 - 9.94T + 67T^{2} \) |
| 71 | \( 1 - 9.14T + 71T^{2} \) |
| 73 | \( 1 - 3.13T + 73T^{2} \) |
| 83 | \( 1 - 0.779T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 9.86T + 97T^{2} \) |
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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
−8.431968371442876360272988830520, −7.23289548791936780601042467538, −6.50928233200405013909896478156, −6.17903917290735867008101277684, −5.55695904871599553055354831057, −4.58833662773704616638909289573, −4.04844123020567487469168163543, −3.68003779348170395772794545458, −1.96109144980644815634535005918, −0.67565296453067578799209420148,
0.67565296453067578799209420148, 1.96109144980644815634535005918, 3.68003779348170395772794545458, 4.04844123020567487469168163543, 4.58833662773704616638909289573, 5.55695904871599553055354831057, 6.17903917290735867008101277684, 6.50928233200405013909896478156, 7.23289548791936780601042467538, 8.431968371442876360272988830520
