| L(s) = 1 | − 2-s + 3-s + 4-s + 1.64·5-s − 6-s − 8-s + 9-s − 1.64·10-s − 4.64·11-s + 12-s + 13-s + 1.64·15-s + 16-s − 1.35·17-s − 18-s + 5·19-s + 1.64·20-s + 4.64·22-s − 7.64·23-s − 24-s − 2.29·25-s − 26-s + 27-s − 6.29·29-s − 1.64·30-s − 7.29·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.736·5-s − 0.408·6-s − 0.353·8-s + 0.333·9-s − 0.520·10-s − 1.40·11-s + 0.288·12-s + 0.277·13-s + 0.424·15-s + 0.250·16-s − 0.328·17-s − 0.235·18-s + 1.14·19-s + 0.368·20-s + 0.990·22-s − 1.59·23-s − 0.204·24-s − 0.458·25-s − 0.196·26-s + 0.192·27-s − 1.16·29-s − 0.300·30-s − 1.30·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 1.64T + 5T^{2} \) |
| 11 | \( 1 + 4.64T + 11T^{2} \) |
| 17 | \( 1 + 1.35T + 17T^{2} \) |
| 19 | \( 1 - 5T + 19T^{2} \) |
| 23 | \( 1 + 7.64T + 23T^{2} \) |
| 29 | \( 1 + 6.29T + 29T^{2} \) |
| 31 | \( 1 + 7.29T + 31T^{2} \) |
| 37 | \( 1 - 0.354T + 37T^{2} \) |
| 41 | \( 1 + 7.64T + 41T^{2} \) |
| 43 | \( 1 - 5.29T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 7.93T + 59T^{2} \) |
| 61 | \( 1 - 3.93T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 5.70T + 71T^{2} \) |
| 73 | \( 1 + 8.35T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 2.70T + 83T^{2} \) |
| 89 | \( 1 - 1.06T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−8.075866292435197277483842580689, −7.61723073235035246849060056607, −6.89985659807353539749792378616, −5.70378190625531471709317989893, −5.51237617004546589237254956116, −4.14316911150518593870444560600, −3.18906377798137353750742715907, −2.28330578840492803100587690073, −1.63302423646835084744839110438, 0,
1.63302423646835084744839110438, 2.28330578840492803100587690073, 3.18906377798137353750742715907, 4.14316911150518593870444560600, 5.51237617004546589237254956116, 5.70378190625531471709317989893, 6.89985659807353539749792378616, 7.61723073235035246849060056607, 8.075866292435197277483842580689
