| L(s) = 1 | − 2·2-s + 4·4-s + 5·5-s + 29.0·7-s − 8·8-s − 10·10-s + 9.76·11-s − 10.6·13-s − 58.1·14-s + 16·16-s − 93.9·17-s − 128.·19-s + 20·20-s − 19.5·22-s − 123.·23-s + 25·25-s + 21.3·26-s + 116.·28-s − 295.·29-s + 156.·31-s − 32·32-s + 187.·34-s + 145.·35-s − 400.·37-s + 257.·38-s − 40·40-s − 99.3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.57·7-s − 0.353·8-s − 0.316·10-s + 0.267·11-s − 0.227·13-s − 1.11·14-s + 0.250·16-s − 1.34·17-s − 1.55·19-s + 0.223·20-s − 0.189·22-s − 1.11·23-s + 0.200·25-s + 0.160·26-s + 0.785·28-s − 1.89·29-s + 0.904·31-s − 0.176·32-s + 0.947·34-s + 0.702·35-s − 1.77·37-s + 1.09·38-s − 0.158·40-s − 0.378·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 - 29.0T + 343T^{2} \) |
| 11 | \( 1 - 9.76T + 1.33e3T^{2} \) |
| 13 | \( 1 + 10.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 128.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 123.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 295.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 400.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 99.3T + 6.89e4T^{2} \) |
| 43 | \( 1 + 30.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 223.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 24.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 589.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 81.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 226.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 216.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 333.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 396.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 473.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 681.T + 9.12e5T^{2} \) |
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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
−9.266372314763588171536681057387, −8.603424938840631967812126758421, −7.933469524646150373380918546744, −6.94390979365128848724572487025, −6.04143183843108051470431690959, −4.94799972125413111262300675163, −4.02130200457284300436999186864, −2.19407212586586882852183991995, −1.70589313136048350216159162725, 0,
1.70589313136048350216159162725, 2.19407212586586882852183991995, 4.02130200457284300436999186864, 4.94799972125413111262300675163, 6.04143183843108051470431690959, 6.94390979365128848724572487025, 7.933469524646150373380918546744, 8.603424938840631967812126758421, 9.266372314763588171536681057387
