| L(s) = 1 | − 2-s + 0.326·3-s + 4-s + 2.05·5-s − 0.326·6-s + 7-s − 8-s − 2.89·9-s − 2.05·10-s + 0.326·12-s + 6.11·13-s − 14-s + 0.673·15-s + 16-s − 0.239·17-s + 2.89·18-s − 19-s + 2.05·20-s + 0.326·21-s − 7.61·23-s − 0.326·24-s − 0.760·25-s − 6.11·26-s − 1.92·27-s + 28-s − 9.90·29-s − 0.673·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.188·3-s + 0.5·4-s + 0.920·5-s − 0.133·6-s + 0.377·7-s − 0.353·8-s − 0.964·9-s − 0.651·10-s + 0.0943·12-s + 1.69·13-s − 0.267·14-s + 0.173·15-s + 0.250·16-s − 0.0580·17-s + 0.681·18-s − 0.229·19-s + 0.460·20-s + 0.0713·21-s − 1.58·23-s − 0.0667·24-s − 0.152·25-s − 1.19·26-s − 0.370·27-s + 0.188·28-s − 1.84·29-s − 0.122·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.326T + 3T^{2} \) |
| 5 | \( 1 - 2.05T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 13 | \( 1 - 6.11T + 13T^{2} \) |
| 17 | \( 1 + 0.239T + 17T^{2} \) |
| 23 | \( 1 + 7.61T + 23T^{2} \) |
| 29 | \( 1 + 9.90T + 29T^{2} \) |
| 31 | \( 1 + 3.80T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 - 6.77T + 41T^{2} \) |
| 43 | \( 1 + 6.97T + 43T^{2} \) |
| 47 | \( 1 - 4.21T + 47T^{2} \) |
| 53 | \( 1 + 8.68T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 6.50T + 61T^{2} \) |
| 67 | \( 1 + 9.28T + 67T^{2} \) |
| 71 | \( 1 + 8.28T + 71T^{2} \) |
| 73 | \( 1 - 5.37T + 73T^{2} \) |
| 79 | \( 1 + 13.2T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−8.130594610897071403329398947361, −7.43290586456537180330379986316, −6.33829326689684011131067476091, −5.90634494275886850279670167207, −5.35025874919038164279144063586, −3.95884234092868216880613699520, −3.27053568894918616204758428957, −2.02719851857888953669281908130, −1.62635814621842171700154043797, 0,
1.62635814621842171700154043797, 2.02719851857888953669281908130, 3.27053568894918616204758428957, 3.95884234092868216880613699520, 5.35025874919038164279144063586, 5.90634494275886850279670167207, 6.33829326689684011131067476091, 7.43290586456537180330379986316, 8.130594610897071403329398947361
