| L(s) = 1 | + 2-s − 1.76·3-s + 4-s + 2.46·5-s − 1.76·6-s − 7-s + 8-s + 0.122·9-s + 2.46·10-s + 4.45·11-s − 1.76·12-s + 6.20·13-s − 14-s − 4.36·15-s + 16-s + 6.32·17-s + 0.122·18-s − 4.82·19-s + 2.46·20-s + 1.76·21-s + 4.45·22-s + 2.00·23-s − 1.76·24-s + 1.09·25-s + 6.20·26-s + 5.08·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.02·3-s + 0.5·4-s + 1.10·5-s − 0.721·6-s − 0.377·7-s + 0.353·8-s + 0.0407·9-s + 0.780·10-s + 1.34·11-s − 0.510·12-s + 1.72·13-s − 0.267·14-s − 1.12·15-s + 0.250·16-s + 1.53·17-s + 0.0287·18-s − 1.10·19-s + 0.552·20-s + 0.385·21-s + 0.950·22-s + 0.417·23-s − 0.360·24-s + 0.218·25-s + 1.21·26-s + 0.978·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.291294921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.291294921\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 1.76T + 3T^{2} \) |
| 5 | \( 1 - 2.46T + 5T^{2} \) |
| 11 | \( 1 - 4.45T + 11T^{2} \) |
| 13 | \( 1 - 6.20T + 13T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 19 | \( 1 + 4.82T + 19T^{2} \) |
| 23 | \( 1 - 2.00T + 23T^{2} \) |
| 29 | \( 1 - 3.40T + 29T^{2} \) |
| 31 | \( 1 + 9.04T + 31T^{2} \) |
| 37 | \( 1 - 6.21T + 37T^{2} \) |
| 41 | \( 1 - 4.17T + 41T^{2} \) |
| 43 | \( 1 - 3.79T + 43T^{2} \) |
| 47 | \( 1 - 3.21T + 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 + 5.71T + 59T^{2} \) |
| 61 | \( 1 + 6.69T + 61T^{2} \) |
| 67 | \( 1 + 0.612T + 67T^{2} \) |
| 71 | \( 1 + 15.3T + 71T^{2} \) |
| 73 | \( 1 - 5.75T + 73T^{2} \) |
| 79 | \( 1 + 3.60T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 11.1T + 89T^{2} \) |
| 97 | \( 1 - 3.22T + 97T^{2} \) |
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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
−7.984427025087755279583556998725, −6.93167981856959984203673042668, −6.24623090397153077582227230533, −5.95400905010274032474137121965, −5.56951150995020424701390937356, −4.48181522588739907874602561522, −3.75100719669093334983164023814, −2.94484443530790948690185609171, −1.68491653717357825061939622529, −0.992526318925146251575749496042,
0.992526318925146251575749496042, 1.68491653717357825061939622529, 2.94484443530790948690185609171, 3.75100719669093334983164023814, 4.48181522588739907874602561522, 5.56951150995020424701390937356, 5.95400905010274032474137121965, 6.24623090397153077582227230533, 6.93167981856959984203673042668, 7.984427025087755279583556998725
