| L(s) = 1 | + 2.61·2-s + 2.17·3-s + 4.85·4-s + 5.70·6-s + 4.94·7-s + 7.47·8-s + 1.74·9-s + 11-s + 10.5·12-s − 4.20·13-s + 12.9·14-s + 9.84·16-s − 6.38·17-s + 4.56·18-s + 19-s + 10.7·21-s + 2.61·22-s + 6.66·23-s + 16.2·24-s − 11.0·26-s − 2.73·27-s + 24.0·28-s − 6.65·29-s − 9.53·31-s + 10.8·32-s + 2.17·33-s − 16.7·34-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 1.25·3-s + 2.42·4-s + 2.32·6-s + 1.87·7-s + 2.64·8-s + 0.581·9-s + 0.301·11-s + 3.05·12-s − 1.16·13-s + 3.46·14-s + 2.46·16-s − 1.54·17-s + 1.07·18-s + 0.229·19-s + 2.35·21-s + 0.558·22-s + 1.38·23-s + 3.32·24-s − 2.15·26-s − 0.526·27-s + 4.53·28-s − 1.23·29-s − 1.71·31-s + 1.91·32-s + 0.379·33-s − 2.86·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.64979084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.64979084\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 3 | \( 1 - 2.17T + 3T^{2} \) |
| 7 | \( 1 - 4.94T + 7T^{2} \) |
| 13 | \( 1 + 4.20T + 13T^{2} \) |
| 17 | \( 1 + 6.38T + 17T^{2} \) |
| 23 | \( 1 - 6.66T + 23T^{2} \) |
| 29 | \( 1 + 6.65T + 29T^{2} \) |
| 31 | \( 1 + 9.53T + 31T^{2} \) |
| 37 | \( 1 + 3.69T + 37T^{2} \) |
| 41 | \( 1 + 1.08T + 41T^{2} \) |
| 43 | \( 1 - 9.58T + 43T^{2} \) |
| 47 | \( 1 - 2.90T + 47T^{2} \) |
| 53 | \( 1 + 6.75T + 53T^{2} \) |
| 59 | \( 1 + 0.613T + 59T^{2} \) |
| 61 | \( 1 - 1.33T + 61T^{2} \) |
| 67 | \( 1 - 0.386T + 67T^{2} \) |
| 71 | \( 1 - 9.38T + 71T^{2} \) |
| 73 | \( 1 + 1.97T + 73T^{2} \) |
| 79 | \( 1 + 6.61T + 79T^{2} \) |
| 83 | \( 1 + 0.100T + 83T^{2} \) |
| 89 | \( 1 - 2.48T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
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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
−7.906290678752296334087030115103, −7.34108004514585695064971243403, −6.92830364128086069344722014866, −5.62056053618688007736868891519, −5.12568298782906921717811313241, −4.42907618623235507687363365084, −3.89009376365356134075823277349, −2.91986899282282655979024683716, −2.16397687271010058425570738793, −1.73423347105317561753912392863,
1.73423347105317561753912392863, 2.16397687271010058425570738793, 2.91986899282282655979024683716, 3.89009376365356134075823277349, 4.42907618623235507687363365084, 5.12568298782906921717811313241, 5.62056053618688007736868891519, 6.92830364128086069344722014866, 7.34108004514585695064971243403, 7.906290678752296334087030115103
