| L(s) = 1 | + 2.74·2-s − 0.876·3-s + 5.54·4-s + 3.27·5-s − 2.40·6-s + 1.12·7-s + 9.72·8-s − 2.23·9-s + 8.99·10-s − 4.85·12-s + 6.68·13-s + 3.08·14-s − 2.87·15-s + 15.6·16-s − 0.812·17-s − 6.13·18-s + 3.99·19-s + 18.1·20-s − 0.984·21-s − 3.36·23-s − 8.51·24-s + 5.73·25-s + 18.3·26-s + 4.58·27-s + 6.22·28-s − 3.04·29-s − 7.88·30-s + ⋯ |
| L(s) = 1 | + 1.94·2-s − 0.505·3-s + 2.77·4-s + 1.46·5-s − 0.982·6-s + 0.424·7-s + 3.43·8-s − 0.744·9-s + 2.84·10-s − 1.40·12-s + 1.85·13-s + 0.824·14-s − 0.741·15-s + 3.90·16-s − 0.197·17-s − 1.44·18-s + 0.916·19-s + 4.05·20-s − 0.214·21-s − 0.702·23-s − 1.73·24-s + 1.14·25-s + 3.59·26-s + 0.882·27-s + 1.17·28-s − 0.565·29-s − 1.43·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5203 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.844226031\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.844226031\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 3 | \( 1 + 0.876T + 3T^{2} \) |
| 5 | \( 1 - 3.27T + 5T^{2} \) |
| 7 | \( 1 - 1.12T + 7T^{2} \) |
| 13 | \( 1 - 6.68T + 13T^{2} \) |
| 17 | \( 1 + 0.812T + 17T^{2} \) |
| 19 | \( 1 - 3.99T + 19T^{2} \) |
| 23 | \( 1 + 3.36T + 23T^{2} \) |
| 29 | \( 1 + 3.04T + 29T^{2} \) |
| 31 | \( 1 + 8.54T + 31T^{2} \) |
| 37 | \( 1 + 6.51T + 37T^{2} \) |
| 41 | \( 1 + 9.92T + 41T^{2} \) |
| 47 | \( 1 + 7.84T + 47T^{2} \) |
| 53 | \( 1 - 2.06T + 53T^{2} \) |
| 59 | \( 1 + 11.2T + 59T^{2} \) |
| 61 | \( 1 - 0.0314T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 5.59T + 71T^{2} \) |
| 73 | \( 1 + 5.17T + 73T^{2} \) |
| 79 | \( 1 - 2.08T + 79T^{2} \) |
| 83 | \( 1 - 7.00T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 7.03T + 97T^{2} \) |
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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
−8.013825078334070544488889774120, −6.95404450092725896801019184342, −6.31504481591853344633329424253, −5.85068805912031128689285002158, −5.37652030615462337578635136987, −4.84035265548133943033437042378, −3.63337763319724593483522912067, −3.19011745462767888728416072587, −1.95142950206119153574253868290, −1.52853765797714209748176727068,
1.52853765797714209748176727068, 1.95142950206119153574253868290, 3.19011745462767888728416072587, 3.63337763319724593483522912067, 4.84035265548133943033437042378, 5.37652030615462337578635136987, 5.85068805912031128689285002158, 6.31504481591853344633329424253, 6.95404450092725896801019184342, 8.013825078334070544488889774120
