| L(s) = 1 | − 2.37·2-s − 3-s + 3.63·4-s − 1.15·5-s + 2.37·6-s − 7-s − 3.88·8-s + 9-s + 2.73·10-s + 2.37·11-s − 3.63·12-s + 4.81·13-s + 2.37·14-s + 1.15·15-s + 1.95·16-s + 0.558·17-s − 2.37·18-s − 6.79·19-s − 4.19·20-s + 21-s − 5.62·22-s + 6.55·23-s + 3.88·24-s − 3.67·25-s − 11.4·26-s − 27-s − 3.63·28-s + ⋯ |
| L(s) = 1 | − 1.67·2-s − 0.577·3-s + 1.81·4-s − 0.515·5-s + 0.969·6-s − 0.377·7-s − 1.37·8-s + 0.333·9-s + 0.865·10-s + 0.714·11-s − 1.05·12-s + 1.33·13-s + 0.634·14-s + 0.297·15-s + 0.489·16-s + 0.135·17-s − 0.559·18-s − 1.55·19-s − 0.937·20-s + 0.218·21-s − 1.20·22-s + 1.36·23-s + 0.793·24-s − 0.734·25-s − 2.24·26-s − 0.192·27-s − 0.687·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 5 | \( 1 + 1.15T + 5T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 - 0.558T + 17T^{2} \) |
| 19 | \( 1 + 6.79T + 19T^{2} \) |
| 23 | \( 1 - 6.55T + 23T^{2} \) |
| 29 | \( 1 - 0.273T + 29T^{2} \) |
| 31 | \( 1 + 0.981T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 3.51T + 41T^{2} \) |
| 43 | \( 1 - 3.95T + 43T^{2} \) |
| 47 | \( 1 + 6.70T + 47T^{2} \) |
| 53 | \( 1 + 7.19T + 53T^{2} \) |
| 59 | \( 1 - 3.36T + 59T^{2} \) |
| 61 | \( 1 + 6.54T + 61T^{2} \) |
| 67 | \( 1 + 9.48T + 67T^{2} \) |
| 71 | \( 1 - 7.84T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 1.60T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 - 0.598T + 97T^{2} \) |
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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
−7.76081225321567683758808256838, −6.72299837009690583880315129669, −6.50829644854546079840231756736, −5.79857378561628577803034050264, −4.54888530963516196230228967920, −3.87508007963200068245925740932, −2.89199522324466779559028170100, −1.73189776742390240924748601792, −0.995243175640523194335856686812, 0,
0.995243175640523194335856686812, 1.73189776742390240924748601792, 2.89199522324466779559028170100, 3.87508007963200068245925740932, 4.54888530963516196230228967920, 5.79857378561628577803034050264, 6.50829644854546079840231756736, 6.72299837009690583880315129669, 7.76081225321567683758808256838
