| L(s) = 1 | − 2-s − 3-s + 4-s − 0.284·5-s + 6-s − 1.57·7-s − 8-s + 9-s + 0.284·10-s + 4.76·11-s − 12-s − 13-s + 1.57·14-s + 0.284·15-s + 16-s + 0.813·17-s − 18-s + 5.65·19-s − 0.284·20-s + 1.57·21-s − 4.76·22-s − 0.125·23-s + 24-s − 4.91·25-s + 26-s − 27-s − 1.57·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.127·5-s + 0.408·6-s − 0.593·7-s − 0.353·8-s + 0.333·9-s + 0.0900·10-s + 1.43·11-s − 0.288·12-s − 0.277·13-s + 0.419·14-s + 0.0735·15-s + 0.250·16-s + 0.197·17-s − 0.235·18-s + 1.29·19-s − 0.0636·20-s + 0.342·21-s − 1.01·22-s − 0.0261·23-s + 0.204·24-s − 0.983·25-s + 0.196·26-s − 0.192·27-s − 0.296·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 0.284T + 5T^{2} \) |
| 7 | \( 1 + 1.57T + 7T^{2} \) |
| 11 | \( 1 - 4.76T + 11T^{2} \) |
| 17 | \( 1 - 0.813T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 0.125T + 23T^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 - 9.11T + 37T^{2} \) |
| 41 | \( 1 - 0.508T + 41T^{2} \) |
| 43 | \( 1 - 1.30T + 43T^{2} \) |
| 47 | \( 1 + 1.48T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 8.72T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 + 14.8T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 + 1.39T + 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 - 7.82T + 89T^{2} \) |
| 97 | \( 1 + 9.35T + 97T^{2} \) |
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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
−7.56008515593580979172762566362, −6.80958827430842132008768414687, −6.18975311551955370010371845799, −5.66039888630740836506285311464, −4.68869450013000127305885086269, −3.76305477451289043372570407186, −3.19095106638186756042874465627, −1.92468346032399858774069431573, −1.11784762162036246495767798716, 0,
1.11784762162036246495767798716, 1.92468346032399858774069431573, 3.19095106638186756042874465627, 3.76305477451289043372570407186, 4.68869450013000127305885086269, 5.66039888630740836506285311464, 6.18975311551955370010371845799, 6.80958827430842132008768414687, 7.56008515593580979172762566362
