| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 4·7-s + 8-s + 9-s − 10-s + 11-s − 12-s − 13-s − 4·14-s + 15-s + 16-s + 2·17-s + 18-s + 4·19-s − 20-s + 4·21-s + 22-s − 24-s + 25-s − 26-s − 27-s − 4·28-s + 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 1.06·14-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.872·21-s + 0.213·22-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.755·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.77763005683217481127179466102, −7.01475079689898501136764801207, −6.51823142610480015605160411372, −5.84321845783311028259783540326, −5.04432061449073492590133386922, −4.25109103242415128686570536188, −3.33456815891444456754221616974, −2.88849744438844658081674728367, −1.33541831418988471616841407860, 0,
1.33541831418988471616841407860, 2.88849744438844658081674728367, 3.33456815891444456754221616974, 4.25109103242415128686570536188, 5.04432061449073492590133386922, 5.84321845783311028259783540326, 6.51823142610480015605160411372, 7.01475079689898501136764801207, 7.77763005683217481127179466102
