| L(s) = 1 | + 2-s + 0.860·3-s + 4-s − 0.340·5-s + 0.860·6-s − 7-s + 8-s − 2.26·9-s − 0.340·10-s + 5.68·11-s + 0.860·12-s + 0.227·13-s − 14-s − 0.293·15-s + 16-s − 8.10·17-s − 2.26·18-s − 4.24·19-s − 0.340·20-s − 0.860·21-s + 5.68·22-s + 2.64·23-s + 0.860·24-s − 4.88·25-s + 0.227·26-s − 4.52·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.496·3-s + 0.5·4-s − 0.152·5-s + 0.351·6-s − 0.377·7-s + 0.353·8-s − 0.753·9-s − 0.107·10-s + 1.71·11-s + 0.248·12-s + 0.0630·13-s − 0.267·14-s − 0.0756·15-s + 0.250·16-s − 1.96·17-s − 0.532·18-s − 0.974·19-s − 0.0761·20-s − 0.187·21-s + 1.21·22-s + 0.552·23-s + 0.175·24-s − 0.976·25-s + 0.0446·26-s − 0.870·27-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.860T + 3T^{2} \) |
| 5 | \( 1 + 0.340T + 5T^{2} \) |
| 11 | \( 1 - 5.68T + 11T^{2} \) |
| 13 | \( 1 - 0.227T + 13T^{2} \) |
| 17 | \( 1 + 8.10T + 17T^{2} \) |
| 19 | \( 1 + 4.24T + 19T^{2} \) |
| 23 | \( 1 - 2.64T + 23T^{2} \) |
| 29 | \( 1 + 3.69T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 + 6.09T + 37T^{2} \) |
| 41 | \( 1 - 3.58T + 41T^{2} \) |
| 43 | \( 1 - 4.72T + 43T^{2} \) |
| 47 | \( 1 + 0.338T + 47T^{2} \) |
| 53 | \( 1 - 8.13T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 8.33T + 61T^{2} \) |
| 67 | \( 1 + 0.319T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 - 4.74T + 73T^{2} \) |
| 79 | \( 1 - 9.18T + 79T^{2} \) |
| 83 | \( 1 + 5.46T + 83T^{2} \) |
| 89 | \( 1 + 4.64T + 89T^{2} \) |
| 97 | \( 1 - 4.79T + 97T^{2} \) |
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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
−7.57166876121010639484216897045, −6.87227227121989927558368908055, −6.26642002562403999978004434771, −5.74039460070307556711737317220, −4.54136225079814610421087585899, −4.03132289817551190575377958455, −3.38457714578875388245394029613, −2.41396492209342804043146836530, −1.69020459461500700772399437520, 0,
1.69020459461500700772399437520, 2.41396492209342804043146836530, 3.38457714578875388245394029613, 4.03132289817551190575377958455, 4.54136225079814610421087585899, 5.74039460070307556711737317220, 6.26642002562403999978004434771, 6.87227227121989927558368908055, 7.57166876121010639484216897045
