| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 4.05·11-s + 12-s + 6.26·13-s − 15-s + 16-s − 3.84·17-s − 18-s − 3.20·19-s − 20-s − 4.05·22-s − 23-s − 24-s − 4·25-s − 6.26·26-s + 27-s − 1.26·29-s + 30-s − 6.47·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.22·11-s + 0.288·12-s + 1.73·13-s − 0.258·15-s + 0.250·16-s − 0.933·17-s − 0.235·18-s − 0.736·19-s − 0.223·20-s − 0.865·22-s − 0.208·23-s − 0.204·24-s − 0.800·25-s − 1.22·26-s + 0.192·27-s − 0.235·29-s + 0.182·30-s − 1.16·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + T + 5T^{2} \) |
| 11 | \( 1 - 4.05T + 11T^{2} \) |
| 13 | \( 1 - 6.26T + 13T^{2} \) |
| 17 | \( 1 + 3.84T + 17T^{2} \) |
| 19 | \( 1 + 3.20T + 19T^{2} \) |
| 29 | \( 1 + 1.26T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 + 8.11T + 43T^{2} \) |
| 47 | \( 1 + 5.47T + 47T^{2} \) |
| 53 | \( 1 + 9.11T + 53T^{2} \) |
| 59 | \( 1 - 6.84T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 12.3T + 71T^{2} \) |
| 73 | \( 1 + 9.05T + 73T^{2} \) |
| 79 | \( 1 - 9.38T + 79T^{2} \) |
| 83 | \( 1 + 0.0593T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−7.86576457439066335474976917331, −6.90771314545499470368033631760, −6.49299806750454744739962905562, −5.76419640329047567185071555930, −4.50671964964244981549216575879, −3.75515039574860594837380228896, −3.34482398020434968063967857584, −1.95178683359132062183817545955, −1.45667312571264659997871643398, 0,
1.45667312571264659997871643398, 1.95178683359132062183817545955, 3.34482398020434968063967857584, 3.75515039574860594837380228896, 4.50671964964244981549216575879, 5.76419640329047567185071555930, 6.49299806750454744739962905562, 6.90771314545499470368033631760, 7.86576457439066335474976917331
