L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 3·7-s − 8-s − 2·9-s + 2·11-s + 12-s + 13-s + 3·14-s + 16-s − 3·17-s + 2·18-s − 19-s − 3·21-s − 2·22-s + 23-s − 24-s − 26-s − 5·27-s − 3·28-s − 5·29-s − 8·31-s − 32-s + 2·33-s + 3·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.603·11-s + 0.288·12-s + 0.277·13-s + 0.801·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 0.229·19-s − 0.654·21-s − 0.426·22-s + 0.208·23-s − 0.204·24-s − 0.196·26-s − 0.962·27-s − 0.566·28-s − 0.928·29-s − 1.43·31-s − 0.176·32-s + 0.348·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−9.400115216603937442308106741391, −8.925019181414506634794762446017, −8.187366436155753750941904167188, −7.08084872197682116999638941334, −6.43803628048625394061789390103, −5.47809446086921756820160858625, −3.84969924867073870114275032751, −3.08844310830142045553246780849, −1.90909315273381969795467705091, 0,
1.90909315273381969795467705091, 3.08844310830142045553246780849, 3.84969924867073870114275032751, 5.47809446086921756820160858625, 6.43803628048625394061789390103, 7.08084872197682116999638941334, 8.187366436155753750941904167188, 8.925019181414506634794762446017, 9.400115216603937442308106741391