| L(s) = 1 | + 2.36e8·3-s + 1.62e12·5-s − 9.39e14·7-s + 5.73e15·9-s − 1.97e18·11-s − 4.34e19·13-s + 3.83e20·15-s + 3.66e20·17-s + 9.20e21·19-s − 2.21e23·21-s + 8.36e23·23-s − 2.79e23·25-s − 1.04e25·27-s − 3.77e25·29-s + 9.23e23·31-s − 4.67e26·33-s − 1.52e27·35-s − 4.08e27·37-s − 1.02e28·39-s + 2.27e28·41-s − 6.98e28·43-s + 9.30e27·45-s + 3.00e29·47-s + 5.04e29·49-s + 8.65e28·51-s + 1.51e30·53-s − 3.20e30·55-s + ⋯ |
| L(s) = 1 | + 1.05·3-s + 0.950·5-s − 1.52·7-s + 0.114·9-s − 1.18·11-s − 1.39·13-s + 1.00·15-s + 0.107·17-s + 0.385·19-s − 1.61·21-s + 1.23·23-s − 0.0960·25-s − 0.934·27-s − 0.966·29-s + 0.00735·31-s − 1.24·33-s − 1.45·35-s − 1.47·37-s − 1.47·39-s + 1.36·41-s − 1.81·43-s + 0.108·45-s + 1.64·47-s + 1.33·49-s + 0.113·51-s + 1.01·53-s − 1.12·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+35/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(18)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{37}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 2.36e8T + 5.00e16T^{2} \) |
| 5 | \( 1 - 1.62e12T + 2.91e24T^{2} \) |
| 7 | \( 1 + 9.39e14T + 3.78e29T^{2} \) |
| 11 | \( 1 + 1.97e18T + 2.81e36T^{2} \) |
| 13 | \( 1 + 4.34e19T + 9.72e38T^{2} \) |
| 17 | \( 1 - 3.66e20T + 1.16e43T^{2} \) |
| 19 | \( 1 - 9.20e21T + 5.70e44T^{2} \) |
| 23 | \( 1 - 8.36e23T + 4.57e47T^{2} \) |
| 29 | \( 1 + 3.77e25T + 1.52e51T^{2} \) |
| 31 | \( 1 - 9.23e23T + 1.57e52T^{2} \) |
| 37 | \( 1 + 4.08e27T + 7.71e54T^{2} \) |
| 41 | \( 1 - 2.27e28T + 2.80e56T^{2} \) |
| 43 | \( 1 + 6.98e28T + 1.48e57T^{2} \) |
| 47 | \( 1 - 3.00e29T + 3.33e58T^{2} \) |
| 53 | \( 1 - 1.51e30T + 2.23e60T^{2} \) |
| 59 | \( 1 + 2.89e30T + 9.54e61T^{2} \) |
| 61 | \( 1 + 2.82e31T + 3.06e62T^{2} \) |
| 67 | \( 1 - 7.29e31T + 8.17e63T^{2} \) |
| 71 | \( 1 - 2.68e32T + 6.22e64T^{2} \) |
| 73 | \( 1 + 2.69e32T + 1.64e65T^{2} \) |
| 79 | \( 1 + 6.90e32T + 2.61e66T^{2} \) |
| 83 | \( 1 + 4.54e33T + 1.47e67T^{2} \) |
| 89 | \( 1 + 5.48e33T + 1.69e68T^{2} \) |
| 97 | \( 1 - 1.63e34T + 3.44e69T^{2} \) |
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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
−15.31073360289981090652532671319, −13.78046311136324036726035846459, −12.78844986063243266000246045441, −10.08489857034891926239279373635, −9.168254892874601745910391238951, −7.29222897457616762554052815458, −5.50821052610249223650900278030, −3.15638181419415468202331382644, −2.30923951838911526524243083161, 0,
2.30923951838911526524243083161, 3.15638181419415468202331382644, 5.50821052610249223650900278030, 7.29222897457616762554052815458, 9.168254892874601745910391238951, 10.08489857034891926239279373635, 12.78844986063243266000246045441, 13.78046311136324036726035846459, 15.31073360289981090652532671319
