L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 3.49·7-s − 8-s + 9-s − 10-s + 2.98·11-s − 12-s + 0.0262·13-s + 3.49·14-s − 15-s + 16-s − 18-s − 2.12·19-s + 20-s + 3.49·21-s − 2.98·22-s + 7.10·23-s + 24-s + 25-s − 0.0262·26-s − 27-s − 3.49·28-s + 6.04·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 1.32·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.899·11-s − 0.288·12-s + 0.00728·13-s + 0.935·14-s − 0.258·15-s + 0.250·16-s − 0.235·18-s − 0.488·19-s + 0.223·20-s + 0.763·21-s − 0.636·22-s + 1.48·23-s + 0.204·24-s + 0.200·25-s − 0.00515·26-s − 0.192·27-s − 0.661·28-s + 1.12·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + 3.49T + 7T^{2} \) |
| 11 | \( 1 - 2.98T + 11T^{2} \) |
| 13 | \( 1 - 0.0262T + 13T^{2} \) |
| 19 | \( 1 + 2.12T + 19T^{2} \) |
| 23 | \( 1 - 7.10T + 23T^{2} \) |
| 29 | \( 1 - 6.04T + 29T^{2} \) |
| 31 | \( 1 + 5.81T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 3.18T + 41T^{2} \) |
| 43 | \( 1 + 3.81T + 43T^{2} \) |
| 47 | \( 1 - 1.22T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 0.962T + 59T^{2} \) |
| 61 | \( 1 + 4.71T + 61T^{2} \) |
| 67 | \( 1 - 9.85T + 67T^{2} \) |
| 71 | \( 1 + 2.43T + 71T^{2} \) |
| 73 | \( 1 - 2.58T + 73T^{2} \) |
| 79 | \( 1 - 3.86T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−7.16191433375020970196366416525, −6.61953404718544163793905008696, −6.43302101009316868189937559244, −5.49128997987905728823730457982, −4.78105619912452096420674501042, −3.63682638444442067473394924652, −3.10169907622335868625845239930, −1.99349006311947519899685098931, −1.07229536848818552211339161383, 0,
1.07229536848818552211339161383, 1.99349006311947519899685098931, 3.10169907622335868625845239930, 3.63682638444442067473394924652, 4.78105619912452096420674501042, 5.49128997987905728823730457982, 6.43302101009316868189937559244, 6.61953404718544163793905008696, 7.16191433375020970196366416525