| L(s) = 1 | + 0.231·2-s − 3-s − 1.94·4-s − 0.231·6-s + 3.28·7-s − 0.913·8-s + 9-s + 4.94·11-s + 1.94·12-s − 2.57·13-s + 0.761·14-s + 3.68·16-s + 2.57·17-s + 0.231·18-s − 1.51·19-s − 3.28·21-s + 1.14·22-s − 3.28·23-s + 0.913·24-s − 0.596·26-s − 27-s − 6.39·28-s + 9.08·29-s + 8.34·31-s + 2.67·32-s − 4.94·33-s + 0.596·34-s + ⋯ |
| L(s) = 1 | + 0.163·2-s − 0.577·3-s − 0.973·4-s − 0.0945·6-s + 1.24·7-s − 0.323·8-s + 0.333·9-s + 1.49·11-s + 0.561·12-s − 0.714·13-s + 0.203·14-s + 0.920·16-s + 0.625·17-s + 0.0545·18-s − 0.347·19-s − 0.717·21-s + 0.244·22-s − 0.685·23-s + 0.186·24-s − 0.117·26-s − 0.192·27-s − 1.20·28-s + 1.68·29-s + 1.49·31-s + 0.473·32-s − 0.861·33-s + 0.102·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.631051536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.631051536\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 47 | \( 1 - T \) |
| good | 2 | \( 1 - 0.231T + 2T^{2} \) |
| 7 | \( 1 - 3.28T + 7T^{2} \) |
| 11 | \( 1 - 4.94T + 11T^{2} \) |
| 13 | \( 1 + 2.57T + 13T^{2} \) |
| 17 | \( 1 - 2.57T + 17T^{2} \) |
| 19 | \( 1 + 1.51T + 19T^{2} \) |
| 23 | \( 1 + 3.28T + 23T^{2} \) |
| 29 | \( 1 - 9.08T + 29T^{2} \) |
| 31 | \( 1 - 8.34T + 31T^{2} \) |
| 37 | \( 1 + 6.94T + 37T^{2} \) |
| 41 | \( 1 + 3.21T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 53 | \( 1 - 3.72T + 53T^{2} \) |
| 59 | \( 1 + 3.22T + 59T^{2} \) |
| 61 | \( 1 - 3.19T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 4.19T + 71T^{2} \) |
| 73 | \( 1 + 6.34T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 - 7.02T + 83T^{2} \) |
| 89 | \( 1 + 8.56T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.501947822994395034506765798783, −8.016275308819472762663260074720, −7.02038437730234252759808375163, −6.22599480511090230785108117147, −5.43085330420990388543422813154, −4.55394910135175666766873036040, −4.37287620542554958585442185351, −3.20710899152383419234799195427, −1.73433913765224209211285074512, −0.818287664042983156403606791815,
0.818287664042983156403606791815, 1.73433913765224209211285074512, 3.20710899152383419234799195427, 4.37287620542554958585442185351, 4.55394910135175666766873036040, 5.43085330420990388543422813154, 6.22599480511090230785108117147, 7.02038437730234252759808375163, 8.016275308819472762663260074720, 8.501947822994395034506765798783
