| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s − 7-s + 3·8-s + 9-s − 10-s + 11-s + 12-s − 13-s + 14-s − 15-s − 16-s + 5·17-s − 18-s − 2·19-s − 20-s + 21-s − 22-s + 6·23-s − 3·24-s − 4·25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s + 1.21·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.218·21-s − 0.213·22-s + 1.25·23-s − 0.612·24-s − 4/5·25-s + 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6699 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6699 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.69022947039618909579281649443, −6.96497618167868947122981480904, −6.34032638944566424941089270505, −5.33243016407764687515559732756, −5.06102991656455580774169755280, −3.98403912365347747454750484432, −3.26345077272434735803425884103, −1.93490498685364787592580197177, −1.09669121276327238709919706514, 0,
1.09669121276327238709919706514, 1.93490498685364787592580197177, 3.26345077272434735803425884103, 3.98403912365347747454750484432, 5.06102991656455580774169755280, 5.33243016407764687515559732756, 6.34032638944566424941089270505, 6.96497618167868947122981480904, 7.69022947039618909579281649443
