L(s) = 1 | + 2-s − 3-s + 4-s + 0.194·5-s − 6-s − 1.95·7-s + 8-s + 9-s + 0.194·10-s − 6.30·11-s − 12-s + 4.81·13-s − 1.95·14-s − 0.194·15-s + 16-s + 3.20·17-s + 18-s − 0.859·19-s + 0.194·20-s + 1.95·21-s − 6.30·22-s − 24-s − 4.96·25-s + 4.81·26-s − 27-s − 1.95·28-s + 5.41·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.0868·5-s − 0.408·6-s − 0.737·7-s + 0.353·8-s + 0.333·9-s + 0.0614·10-s − 1.90·11-s − 0.288·12-s + 1.33·13-s − 0.521·14-s − 0.0501·15-s + 0.250·16-s + 0.778·17-s + 0.235·18-s − 0.197·19-s + 0.0434·20-s + 0.425·21-s − 1.34·22-s − 0.204·24-s − 0.992·25-s + 0.943·26-s − 0.192·27-s − 0.368·28-s + 1.00·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3174 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - 0.194T + 5T^{2} \) |
| 7 | \( 1 + 1.95T + 7T^{2} \) |
| 11 | \( 1 + 6.30T + 11T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 - 3.20T + 17T^{2} \) |
| 19 | \( 1 + 0.859T + 19T^{2} \) |
| 29 | \( 1 - 5.41T + 29T^{2} \) |
| 31 | \( 1 + 1.01T + 31T^{2} \) |
| 37 | \( 1 + 6.89T + 37T^{2} \) |
| 41 | \( 1 + 0.972T + 41T^{2} \) |
| 43 | \( 1 - 5.14T + 43T^{2} \) |
| 47 | \( 1 + 7.41T + 47T^{2} \) |
| 53 | \( 1 + 2.00T + 53T^{2} \) |
| 59 | \( 1 + 8.60T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 7.09T + 67T^{2} \) |
| 71 | \( 1 - 1.10T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 1.12T + 79T^{2} \) |
| 83 | \( 1 + 4.00T + 83T^{2} \) |
| 89 | \( 1 - 9.55T + 89T^{2} \) |
| 97 | \( 1 - 1.12T + 97T^{2} \) |
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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
−8.055106754778349446268040274499, −7.53647702861701262053637216249, −6.48087009449749395762951865318, −5.96265237040450019433716898207, −5.33472292588361027377155374076, −4.52922289329819428087492871827, −3.46911182979740236116371631095, −2.84493016674768373525730025019, −1.55956949714434538645302076800, 0,
1.55956949714434538645302076800, 2.84493016674768373525730025019, 3.46911182979740236116371631095, 4.52922289329819428087492871827, 5.33472292588361027377155374076, 5.96265237040450019433716898207, 6.48087009449749395762951865318, 7.53647702861701262053637216249, 8.055106754778349446268040274499