| L(s) = 1 | − 2.16·2-s + 3-s + 2.66·4-s − 2.16·6-s + 3.16·7-s − 1.44·8-s + 9-s + 1.53·11-s + 2.66·12-s + 5.24·13-s − 6.82·14-s − 2.21·16-s − 1.29·17-s − 2.16·18-s + 5.44·19-s + 3.16·21-s − 3.32·22-s + 6.44·23-s − 1.44·24-s − 11.3·26-s + 27-s + 8.43·28-s − 2.36·29-s − 4.46·31-s + 7.67·32-s + 1.53·33-s + 2.78·34-s + ⋯ |
| L(s) = 1 | − 1.52·2-s + 0.577·3-s + 1.33·4-s − 0.882·6-s + 1.19·7-s − 0.510·8-s + 0.333·9-s + 0.463·11-s + 0.770·12-s + 1.45·13-s − 1.82·14-s − 0.554·16-s − 0.313·17-s − 0.509·18-s + 1.24·19-s + 0.689·21-s − 0.708·22-s + 1.34·23-s − 0.294·24-s − 2.22·26-s + 0.192·27-s + 1.59·28-s − 0.438·29-s − 0.802·31-s + 1.35·32-s + 0.267·33-s + 0.478·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.381977212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.381977212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.16T + 2T^{2} \) |
| 7 | \( 1 - 3.16T + 7T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 - 5.24T + 13T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 19 | \( 1 - 5.44T + 19T^{2} \) |
| 23 | \( 1 - 6.44T + 23T^{2} \) |
| 29 | \( 1 + 2.36T + 29T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 + 8.53T + 41T^{2} \) |
| 43 | \( 1 - 8.48T + 43T^{2} \) |
| 47 | \( 1 + 0.753T + 47T^{2} \) |
| 53 | \( 1 + 9.74T + 53T^{2} \) |
| 59 | \( 1 - 4.11T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 1.89T + 67T^{2} \) |
| 71 | \( 1 + 0.0708T + 71T^{2} \) |
| 73 | \( 1 + 4.01T + 73T^{2} \) |
| 79 | \( 1 - 1.61T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 7.27T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−9.056890780296418819570891340119, −8.610604609591551071981622707648, −7.84119533213878098710510987125, −7.32743387245155365468124033760, −6.39404754947711846220362408821, −5.21074788867332622963509700254, −4.17592482584135340463122380485, −3.04143014765733930548069180734, −1.70177323033836171511583531514, −1.12143483415779966301937224130,
1.12143483415779966301937224130, 1.70177323033836171511583531514, 3.04143014765733930548069180734, 4.17592482584135340463122380485, 5.21074788867332622963509700254, 6.39404754947711846220362408821, 7.32743387245155365468124033760, 7.84119533213878098710510987125, 8.610604609591551071981622707648, 9.056890780296418819570891340119
