L(s) = 1 | − 2.00·3-s − 2.46·5-s + 4.06·7-s + 1.00·9-s − 3.25·11-s + 4.74·13-s + 4.93·15-s − 4.01·17-s + 19-s − 8.14·21-s − 2.85·23-s + 1.07·25-s + 3.98·27-s − 3.94·29-s + 2.73·31-s + 6.50·33-s − 10.0·35-s + 2.47·37-s − 9.50·39-s + 7.55·41-s + 2.10·43-s − 2.48·45-s − 12.7·47-s + 9.53·49-s + 8.03·51-s + 13.6·53-s + 8.01·55-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1.10·5-s + 1.53·7-s + 0.336·9-s − 0.980·11-s + 1.31·13-s + 1.27·15-s − 0.973·17-s + 0.229·19-s − 1.77·21-s − 0.596·23-s + 0.214·25-s + 0.767·27-s − 0.731·29-s + 0.491·31-s + 1.13·33-s − 1.69·35-s + 0.406·37-s − 1.52·39-s + 1.17·41-s + 0.321·43-s − 0.370·45-s − 1.85·47-s + 1.36·49-s + 1.12·51-s + 1.87·53-s + 1.08·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 + 2.00T + 3T^{2} \) |
| 5 | \( 1 + 2.46T + 5T^{2} \) |
| 7 | \( 1 - 4.06T + 7T^{2} \) |
| 11 | \( 1 + 3.25T + 11T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 17 | \( 1 + 4.01T + 17T^{2} \) |
| 23 | \( 1 + 2.85T + 23T^{2} \) |
| 29 | \( 1 + 3.94T + 29T^{2} \) |
| 31 | \( 1 - 2.73T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 - 7.55T + 41T^{2} \) |
| 43 | \( 1 - 2.10T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 6.69T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 2.77T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 9.87T + 73T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 1.17T + 89T^{2} \) |
| 97 | \( 1 - 7.33e - 5T + 97T^{2} \) |
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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
−7.67587737353842410458465607834, −7.18212351453145351040777473355, −6.01176648772706416259023530179, −5.71277308471914527957644598632, −4.60678834068399516143413002902, −4.48003983246692912011520151682, −3.39450913559179770710691433326, −2.17231974017791806845221768486, −1.07134899847730775915191384810, 0,
1.07134899847730775915191384810, 2.17231974017791806845221768486, 3.39450913559179770710691433326, 4.48003983246692912011520151682, 4.60678834068399516143413002902, 5.71277308471914527957644598632, 6.01176648772706416259023530179, 7.18212351453145351040777473355, 7.67587737353842410458465607834