| L(s) = 1 | + 3·3-s + 12.8·5-s + 9·9-s − 36.8·11-s − 87.1·13-s + 38.4·15-s − 102.·17-s − 95.8·19-s − 96·23-s + 39.4·25-s + 27·27-s − 212.·29-s + 159.·31-s − 110.·33-s + 128.·37-s − 261.·39-s + 298.·41-s − 33.3·43-s + 115.·45-s − 271.·47-s − 307.·51-s + 448.·53-s − 472.·55-s − 287.·57-s + 668.·59-s + 243.·61-s − 1.11e3·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.14·5-s + 0.333·9-s − 1.00·11-s − 1.85·13-s + 0.662·15-s − 1.46·17-s − 1.15·19-s − 0.870·23-s + 0.315·25-s + 0.192·27-s − 1.35·29-s + 0.922·31-s − 0.582·33-s + 0.571·37-s − 1.07·39-s + 1.13·41-s − 0.118·43-s + 0.382·45-s − 0.841·47-s − 0.845·51-s + 1.16·53-s − 1.15·55-s − 0.668·57-s + 1.47·59-s + 0.511·61-s − 2.13·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 12.8T + 125T^{2} \) |
| 11 | \( 1 + 36.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 87.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 95.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 96T + 1.21e4T^{2} \) |
| 29 | \( 1 + 212.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 159.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 128.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 298.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 33.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 271.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 448.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 668.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 243.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 335.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 339.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 918.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 136.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 287.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 161.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 182.T + 9.12e5T^{2} \) |
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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
−9.873506667717638446185165015147, −9.106784528495630553367999833616, −8.116516116234595807110556206609, −7.20840183837775313505164658179, −6.21011590407670477029483368486, −5.17268696859433068105000369694, −4.22974700376419267558736971361, −2.39002319408523165345926408020, −2.23559698147014971048152971589, 0,
2.23559698147014971048152971589, 2.39002319408523165345926408020, 4.22974700376419267558736971361, 5.17268696859433068105000369694, 6.21011590407670477029483368486, 7.20840183837775313505164658179, 8.116516116234595807110556206609, 9.106784528495630553367999833616, 9.873506667717638446185165015147
