| L(s) = 1 | + 3·2-s − 3-s + 4-s − 9·5-s − 3·6-s + 15·7-s − 21·8-s − 26·9-s − 27·10-s − 48·11-s − 12-s + 45·14-s + 9·15-s − 71·16-s + 45·17-s − 78·18-s + 6·19-s − 9·20-s − 15·21-s − 144·22-s − 162·23-s + 21·24-s − 44·25-s + 53·27-s + 15·28-s − 144·29-s + 27·30-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.192·3-s + 1/8·4-s − 0.804·5-s − 0.204·6-s + 0.809·7-s − 0.928·8-s − 0.962·9-s − 0.853·10-s − 1.31·11-s − 0.0240·12-s + 0.859·14-s + 0.154·15-s − 1.10·16-s + 0.642·17-s − 1.02·18-s + 0.0724·19-s − 0.100·20-s − 0.155·21-s − 1.39·22-s − 1.46·23-s + 0.178·24-s − 0.351·25-s + 0.377·27-s + 0.101·28-s − 0.922·29-s + 0.164·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 3 | \( 1 + T + p^{3} T^{2} \) |
| 5 | \( 1 + 9 T + p^{3} T^{2} \) |
| 7 | \( 1 - 15 T + p^{3} T^{2} \) |
| 11 | \( 1 + 48 T + p^{3} T^{2} \) |
| 17 | \( 1 - 45 T + p^{3} T^{2} \) |
| 19 | \( 1 - 6 T + p^{3} T^{2} \) |
| 23 | \( 1 + 162 T + p^{3} T^{2} \) |
| 29 | \( 1 + 144 T + p^{3} T^{2} \) |
| 31 | \( 1 - 264 T + p^{3} T^{2} \) |
| 37 | \( 1 - 303 T + p^{3} T^{2} \) |
| 41 | \( 1 + 192 T + p^{3} T^{2} \) |
| 43 | \( 1 - 97 T + p^{3} T^{2} \) |
| 47 | \( 1 - 111 T + p^{3} T^{2} \) |
| 53 | \( 1 + 414 T + p^{3} T^{2} \) |
| 59 | \( 1 - 522 T + p^{3} T^{2} \) |
| 61 | \( 1 - 376 T + p^{3} T^{2} \) |
| 67 | \( 1 + 36 T + p^{3} T^{2} \) |
| 71 | \( 1 - 357 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1098 T + p^{3} T^{2} \) |
| 79 | \( 1 + 830 T + p^{3} T^{2} \) |
| 83 | \( 1 + 438 T + p^{3} T^{2} \) |
| 89 | \( 1 + 438 T + p^{3} T^{2} \) |
| 97 | \( 1 + 852 T + p^{3} T^{2} \) |
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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
−11.80563324700555214138807167126, −11.35589695200246032151917693161, −9.979535769891282763047543805064, −8.383878464962703217927453259679, −7.79309974919012910709533416278, −5.98382682438056581135524175595, −5.15688395128338394749027627544, −4.07013122221990655746043641957, −2.73321939390524090745927232414, 0,
2.73321939390524090745927232414, 4.07013122221990655746043641957, 5.15688395128338394749027627544, 5.98382682438056581135524175595, 7.79309974919012910709533416278, 8.383878464962703217927453259679, 9.979535769891282763047543805064, 11.35589695200246032151917693161, 11.80563324700555214138807167126
