| L(s) = 1 | − 4.84·2-s − 9.58·3-s + 15.4·4-s + 46.3·6-s − 36.0·8-s + 64.7·9-s + 62.1·11-s − 147.·12-s + 14.0·13-s + 50.9·16-s + 63.5·17-s − 313.·18-s − 48.7·19-s − 301.·22-s + 99.3·23-s + 345.·24-s − 68.2·26-s − 362.·27-s − 69.0·29-s + 9.68·31-s + 41.7·32-s − 595.·33-s − 307.·34-s + 1.00e3·36-s − 240.·37-s + 235.·38-s − 135.·39-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 1.84·3-s + 1.93·4-s + 3.15·6-s − 1.59·8-s + 2.39·9-s + 1.70·11-s − 3.55·12-s + 0.300·13-s + 0.795·16-s + 0.906·17-s − 4.10·18-s − 0.588·19-s − 2.91·22-s + 0.900·23-s + 2.93·24-s − 0.514·26-s − 2.58·27-s − 0.442·29-s + 0.0561·31-s + 0.230·32-s − 3.14·33-s − 1.55·34-s + 4.63·36-s − 1.06·37-s + 1.00·38-s − 0.554·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.84T + 8T^{2} \) |
| 3 | \( 1 + 9.58T + 27T^{2} \) |
| 11 | \( 1 - 62.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 14.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 48.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 99.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 69.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 9.68T + 2.97e4T^{2} \) |
| 37 | \( 1 + 240.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 335.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 51.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 451.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 180.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 268.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 323.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 541.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 161.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 305.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 504.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 513.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 543.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.86e3T + 9.12e5T^{2} \) |
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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
−9.124922058502696144109531335418, −8.234619366329011785141424004006, −7.03472901752874009035132641081, −6.73699440912720794576766934268, −5.94199711432685700350682523443, −4.90469260113944826905754662155, −3.67396005452955038873843986799, −1.64558723118750688615709789919, −1.06167432870786508679043778622, 0,
1.06167432870786508679043778622, 1.64558723118750688615709789919, 3.67396005452955038873843986799, 4.90469260113944826905754662155, 5.94199711432685700350682523443, 6.73699440912720794576766934268, 7.03472901752874009035132641081, 8.234619366329011785141424004006, 9.124922058502696144109531335418
