| L(s) = 1 | + 10.3·3-s − 9.61·5-s + 11.2·7-s − 136.·9-s − 268.·11-s + 306.·13-s − 99.4·15-s − 2.22e3·17-s − 361·19-s + 116.·21-s + 2.98e3·23-s − 3.03e3·25-s − 3.92e3·27-s + 4.06e3·29-s − 7.46e3·31-s − 2.77e3·33-s − 108.·35-s − 1.19e4·37-s + 3.16e3·39-s + 1.43e4·41-s + 1.08e4·43-s + 1.30e3·45-s − 2.18e4·47-s − 1.66e4·49-s − 2.30e4·51-s − 1.27e4·53-s + 2.58e3·55-s + ⋯ |
| L(s) = 1 | + 0.663·3-s − 0.172·5-s + 0.0868·7-s − 0.559·9-s − 0.669·11-s + 0.502·13-s − 0.114·15-s − 1.86·17-s − 0.229·19-s + 0.0576·21-s + 1.17·23-s − 0.970·25-s − 1.03·27-s + 0.897·29-s − 1.39·31-s − 0.444·33-s − 0.0149·35-s − 1.43·37-s + 0.333·39-s + 1.33·41-s + 0.898·43-s + 0.0963·45-s − 1.44·47-s − 0.992·49-s − 1.23·51-s − 0.622·53-s + 0.115·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + 361T \) |
| good | 3 | \( 1 - 10.3T + 243T^{2} \) |
| 5 | \( 1 + 9.61T + 3.12e3T^{2} \) |
| 7 | \( 1 - 11.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 268.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 306.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.22e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 2.98e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.06e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.46e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.19e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.43e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.08e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.18e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.27e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.12e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.65e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.27e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.11e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.87e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.06e5T + 8.58e9T^{2} \) |
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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
−11.38389709421297545566123476826, −10.73517529014368707838059210694, −9.218214583467328354679144258885, −8.559370862744152039380243915981, −7.48400233940820430514663829601, −6.17052421478278977402228492169, −4.75239562406153369864284287763, −3.32996787032648998185880383968, −2.08679920860777987667351367412, 0,
2.08679920860777987667351367412, 3.32996787032648998185880383968, 4.75239562406153369864284287763, 6.17052421478278977402228492169, 7.48400233940820430514663829601, 8.559370862744152039380243915981, 9.218214583467328354679144258885, 10.73517529014368707838059210694, 11.38389709421297545566123476826
