| L(s) = 1 | + 8·2-s + 64·4-s − 165·5-s − 508·7-s + 512·8-s − 1.32e3·10-s + 3.02e3·11-s + 5.03e3·13-s − 4.06e3·14-s + 4.09e3·16-s − 3.18e3·17-s + 1.50e3·19-s − 1.05e4·20-s + 2.41e4·22-s − 7.56e4·23-s − 5.09e4·25-s + 4.03e4·26-s − 3.25e4·28-s − 8.26e4·29-s − 1.74e5·31-s + 3.27e4·32-s − 2.55e4·34-s + 8.38e4·35-s − 3.23e5·37-s + 1.20e4·38-s − 8.44e4·40-s − 3.08e5·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.590·5-s − 0.559·7-s + 0.353·8-s − 0.417·10-s + 0.685·11-s + 0.636·13-s − 0.395·14-s + 1/4·16-s − 0.157·17-s + 0.0504·19-s − 0.295·20-s + 0.484·22-s − 1.29·23-s − 0.651·25-s + 0.449·26-s − 0.279·28-s − 0.629·29-s − 1.05·31-s + 0.176·32-s − 0.111·34-s + 0.330·35-s − 1.05·37-s + 0.0356·38-s − 0.208·40-s − 0.698·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{3} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 33 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 508 T + p^{7} T^{2} \) |
| 11 | \( 1 - 3024 T + p^{7} T^{2} \) |
| 13 | \( 1 - 5039 T + p^{7} T^{2} \) |
| 17 | \( 1 + 3189 T + p^{7} T^{2} \) |
| 19 | \( 1 - 1508 T + p^{7} T^{2} \) |
| 23 | \( 1 + 75600 T + p^{7} T^{2} \) |
| 29 | \( 1 + 82665 T + p^{7} T^{2} \) |
| 31 | \( 1 + 174892 T + p^{7} T^{2} \) |
| 37 | \( 1 + 323569 T + p^{7} T^{2} \) |
| 41 | \( 1 + 308118 T + p^{7} T^{2} \) |
| 43 | \( 1 - 336680 T + p^{7} T^{2} \) |
| 47 | \( 1 + 383196 T + p^{7} T^{2} \) |
| 53 | \( 1 - 760206 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2225664 T + p^{7} T^{2} \) |
| 61 | \( 1 - 2244815 T + p^{7} T^{2} \) |
| 67 | \( 1 - 1473188 T + p^{7} T^{2} \) |
| 71 | \( 1 + 5006892 T + p^{7} T^{2} \) |
| 73 | \( 1 + 5898301 T + p^{7} T^{2} \) |
| 79 | \( 1 - 7028768 T + p^{7} T^{2} \) |
| 83 | \( 1 + 2651196 T + p^{7} T^{2} \) |
| 89 | \( 1 + 6770901 T + p^{7} T^{2} \) |
| 97 | \( 1 - 16176386 T + p^{7} T^{2} \) |
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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
−11.34043325297840783623196539298, −10.18069604097718888816905520926, −8.971694411553695824216425530445, −7.74597652412341814988104030203, −6.64173360325397727413371338916, −5.66037360169780218180545218815, −4.14021563316544553018951225009, −3.41345366030187447043885596084, −1.75635296670456043562076078548, 0,
1.75635296670456043562076078548, 3.41345366030187447043885596084, 4.14021563316544553018951225009, 5.66037360169780218180545218815, 6.64173360325397727413371338916, 7.74597652412341814988104030203, 8.971694411553695824216425530445, 10.18069604097718888816905520926, 11.34043325297840783623196539298
