| L(s) = 1 | + 2-s − 31·4-s + 92·5-s + 26·7-s − 63·8-s + 92·10-s + 692·13-s + 26·14-s + 929·16-s − 1.44e3·17-s − 2.16e3·19-s − 2.85e3·20-s + 1.58e3·23-s + 5.33e3·25-s + 692·26-s − 806·28-s − 5.52e3·29-s + 4.79e3·31-s + 2.94e3·32-s − 1.44e3·34-s + 2.39e3·35-s − 1.01e4·37-s − 2.16e3·38-s − 5.79e3·40-s − 1.06e4·41-s − 8.58e3·43-s + 1.58e3·46-s + ⋯ |
| L(s) = 1 | + 0.176·2-s − 0.968·4-s + 1.64·5-s + 0.200·7-s − 0.348·8-s + 0.290·10-s + 1.13·13-s + 0.0354·14-s + 0.907·16-s − 1.21·17-s − 1.37·19-s − 1.59·20-s + 0.623·23-s + 1.70·25-s + 0.200·26-s − 0.194·28-s − 1.22·29-s + 0.895·31-s + 0.508·32-s − 0.213·34-s + 0.330·35-s − 1.22·37-s − 0.242·38-s − 0.572·40-s − 0.986·41-s − 0.707·43-s + 0.110·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p^{5} T^{2} \) |
| 5 | \( 1 - 92 T + p^{5} T^{2} \) |
| 7 | \( 1 - 26 T + p^{5} T^{2} \) |
| 13 | \( 1 - 692 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1442 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2160 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1582 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5526 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4792 T + p^{5} T^{2} \) |
| 37 | \( 1 + 10194 T + p^{5} T^{2} \) |
| 41 | \( 1 + 10622 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8580 T + p^{5} T^{2} \) |
| 47 | \( 1 - 2362 T + p^{5} T^{2} \) |
| 53 | \( 1 - 30804 T + p^{5} T^{2} \) |
| 59 | \( 1 + 6416 T + p^{5} T^{2} \) |
| 61 | \( 1 + 42096 T + p^{5} T^{2} \) |
| 67 | \( 1 + 28444 T + p^{5} T^{2} \) |
| 71 | \( 1 + 45690 T + p^{5} T^{2} \) |
| 73 | \( 1 - 18374 T + p^{5} T^{2} \) |
| 79 | \( 1 - 105214 T + p^{5} T^{2} \) |
| 83 | \( 1 - 62292 T + p^{5} T^{2} \) |
| 89 | \( 1 - 72246 T + p^{5} T^{2} \) |
| 97 | \( 1 - 79262 T + p^{5} T^{2} \) |
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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
−8.923175376994406729860227652574, −8.242038902215030216475039715979, −6.70469353508610614340715197059, −6.13869311776910340336655192583, −5.28365118510940846058795773294, −4.52519346929470800875044958260, −3.45509689143640689224867048442, −2.17474754420813958625739413361, −1.35502567712453886380976830967, 0,
1.35502567712453886380976830967, 2.17474754420813958625739413361, 3.45509689143640689224867048442, 4.52519346929470800875044958260, 5.28365118510940846058795773294, 6.13869311776910340336655192583, 6.70469353508610614340715197059, 8.242038902215030216475039715979, 8.923175376994406729860227652574
