| L(s) = 1 | − 3.07·3-s − 2·4-s + 0.618·7-s + 6.47·9-s − 5.85·11-s + 6.15·12-s + 3.07·13-s + 4·16-s + 2.85·17-s − 1.90·21-s − 5.47·23-s − 10.6·27-s − 1.23·28-s − 3.80·29-s − 1.62·31-s + 18.0·33-s − 12.9·36-s + 8.33·37-s − 9.47·39-s − 11.5·41-s + 7.38·43-s + 11.7·44-s − 4.70·47-s − 12.3·48-s − 6.61·49-s − 8.78·51-s − 6.15·52-s + ⋯ |
| L(s) = 1 | − 1.77·3-s − 4-s + 0.233·7-s + 2.15·9-s − 1.76·11-s + 1.77·12-s + 0.853·13-s + 16-s + 0.692·17-s − 0.415·21-s − 1.14·23-s − 2.05·27-s − 0.233·28-s − 0.706·29-s − 0.291·31-s + 3.13·33-s − 2.15·36-s + 1.37·37-s − 1.51·39-s − 1.80·41-s + 1.12·43-s + 1.76·44-s − 0.686·47-s − 1.77·48-s − 0.945·49-s − 1.23·51-s − 0.853·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 2T^{2} \) |
| 3 | \( 1 + 3.07T + 3T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 + 5.85T + 11T^{2} \) |
| 13 | \( 1 - 3.07T + 13T^{2} \) |
| 17 | \( 1 - 2.85T + 17T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 + 3.80T + 29T^{2} \) |
| 31 | \( 1 + 1.62T + 31T^{2} \) |
| 37 | \( 1 - 8.33T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 - 7.38T + 43T^{2} \) |
| 47 | \( 1 + 4.70T + 47T^{2} \) |
| 53 | \( 1 - 5.15T + 53T^{2} \) |
| 59 | \( 1 - 4.25T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 + 7.33T + 67T^{2} \) |
| 71 | \( 1 - 0.171T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 + 10.7T + 83T^{2} \) |
| 89 | \( 1 - 8.22T + 89T^{2} \) |
| 97 | \( 1 - 8.33T + 97T^{2} \) |
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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
−7.57123743971966380194233071568, −6.39136282524663150988450619307, −5.84940197768482474769922887034, −5.32548001740645601111780297404, −4.86264095157405267887580820944, −4.12353952679138794097850491691, −3.30025791707696752984701861475, −1.88634410024238724564936689920, −0.815525338573457551274804757173, 0,
0.815525338573457551274804757173, 1.88634410024238724564936689920, 3.30025791707696752984701861475, 4.12353952679138794097850491691, 4.86264095157405267887580820944, 5.32548001740645601111780297404, 5.84940197768482474769922887034, 6.39136282524663150988450619307, 7.57123743971966380194233071568
