| L(s) = 1 | − 1.25·2-s − 6.43·4-s − 5.67·7-s + 18.0·8-s − 21.2·11-s + 40.9·13-s + 7.09·14-s + 28.8·16-s − 59.1·17-s + 21.8·19-s + 26.5·22-s + 98.7·23-s − 51.2·26-s + 36.4·28-s + 159.·29-s − 69.4·31-s − 180.·32-s + 74.0·34-s − 235.·37-s − 27.3·38-s + 491.·41-s − 95.3·43-s + 136.·44-s − 123.·46-s + 548.·47-s − 310.·49-s − 263.·52-s + ⋯ |
| L(s) = 1 | − 0.442·2-s − 0.804·4-s − 0.306·7-s + 0.798·8-s − 0.582·11-s + 0.873·13-s + 0.135·14-s + 0.451·16-s − 0.844·17-s + 0.264·19-s + 0.257·22-s + 0.895·23-s − 0.386·26-s + 0.246·28-s + 1.02·29-s − 0.402·31-s − 0.997·32-s + 0.373·34-s − 1.04·37-s − 0.116·38-s + 1.87·41-s − 0.338·43-s + 0.468·44-s − 0.396·46-s + 1.70·47-s − 0.906·49-s − 0.702·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 1.25T + 8T^{2} \) |
| 7 | \( 1 + 5.67T + 343T^{2} \) |
| 11 | \( 1 + 21.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 40.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 59.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 21.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 98.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 159.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 69.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 235.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 491.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 95.3T + 7.95e4T^{2} \) |
| 47 | \( 1 - 548.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 509.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 741.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 387.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.06e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 508.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 914.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 925.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 708.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 273.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.45e3T + 9.12e5T^{2} \) |
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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
−9.474861324537127375555735951954, −8.903428681833268758505127404426, −8.106266530980269323259966588514, −7.17292406268072486055057517997, −6.06468636022131345938700644015, −5.01635413763030206225762878796, −4.10315780838093171198960157607, −2.90830100011878057352981219804, −1.28118990164797353014875740776, 0,
1.28118990164797353014875740776, 2.90830100011878057352981219804, 4.10315780838093171198960157607, 5.01635413763030206225762878796, 6.06468636022131345938700644015, 7.17292406268072486055057517997, 8.106266530980269323259966588514, 8.903428681833268758505127404426, 9.474861324537127375555735951954
