| L(s) = 1 | − 4.78·3-s + 12.0·5-s + 4.33·7-s − 4.11·9-s − 30.0·11-s + 21.0·13-s − 57.8·15-s − 29.7·19-s − 20.7·21-s + 88.1·23-s + 21.3·25-s + 148.·27-s − 27.2·29-s + 88.0·31-s + 143.·33-s + 52.4·35-s − 244.·37-s − 100.·39-s + 86.2·41-s + 385.·43-s − 49.7·45-s − 35.9·47-s − 324.·49-s − 220.·53-s − 363.·55-s + 142.·57-s + 234.·59-s + ⋯ |
| L(s) = 1 | − 0.920·3-s + 1.08·5-s + 0.234·7-s − 0.152·9-s − 0.822·11-s + 0.448·13-s − 0.996·15-s − 0.359·19-s − 0.215·21-s + 0.798·23-s + 0.171·25-s + 1.06·27-s − 0.174·29-s + 0.509·31-s + 0.757·33-s + 0.253·35-s − 1.08·37-s − 0.413·39-s + 0.328·41-s + 1.36·43-s − 0.164·45-s − 0.111·47-s − 0.945·49-s − 0.571·53-s − 0.889·55-s + 0.330·57-s + 0.517·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 4.78T + 27T^{2} \) |
| 5 | \( 1 - 12.0T + 125T^{2} \) |
| 7 | \( 1 - 4.33T + 343T^{2} \) |
| 11 | \( 1 + 30.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.0T + 2.19e3T^{2} \) |
| 19 | \( 1 + 29.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 88.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 27.2T + 2.43e4T^{2} \) |
| 31 | \( 1 - 88.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 244.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 86.2T + 6.89e4T^{2} \) |
| 43 | \( 1 - 385.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 35.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 220.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 234.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 892.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 846.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 391.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 204.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 23.0T + 4.93e5T^{2} \) |
| 83 | \( 1 + 742.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3T + 9.12e5T^{2} \) |
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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
−8.350586570551355354669133732972, −7.38712829941533311958697553232, −6.43874744248048241976670504961, −5.87871659713579865642160818881, −5.26520127633555988213821749436, −4.54984571863877452872823395154, −3.16883082025344807282551810843, −2.22906943455200820865992974434, −1.17207225578257355354517128765, 0,
1.17207225578257355354517128765, 2.22906943455200820865992974434, 3.16883082025344807282551810843, 4.54984571863877452872823395154, 5.26520127633555988213821749436, 5.87871659713579865642160818881, 6.43874744248048241976670504961, 7.38712829941533311958697553232, 8.350586570551355354669133732972
