| L(s) = 1 | + 1.19·2-s − 3.04·3-s − 0.581·4-s + 5-s − 3.62·6-s − 0.609·7-s − 3.07·8-s + 6.28·9-s + 1.19·10-s + 4.48·11-s + 1.77·12-s − 4.43·13-s − 0.725·14-s − 3.04·15-s − 2.49·16-s + 2.90·17-s + 7.48·18-s − 0.581·20-s + 1.85·21-s + 5.34·22-s − 2.84·23-s + 9.36·24-s + 25-s − 5.28·26-s − 10.0·27-s + 0.354·28-s − 1.11·29-s + ⋯ |
| L(s) = 1 | + 0.842·2-s − 1.75·3-s − 0.290·4-s + 0.447·5-s − 1.48·6-s − 0.230·7-s − 1.08·8-s + 2.09·9-s + 0.376·10-s + 1.35·11-s + 0.511·12-s − 1.23·13-s − 0.193·14-s − 0.786·15-s − 0.624·16-s + 0.704·17-s + 1.76·18-s − 0.130·20-s + 0.405·21-s + 1.13·22-s − 0.593·23-s + 1.91·24-s + 0.200·25-s − 1.03·26-s − 1.92·27-s + 0.0669·28-s − 0.207·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 1.19T + 2T^{2} \) |
| 3 | \( 1 + 3.04T + 3T^{2} \) |
| 7 | \( 1 + 0.609T + 7T^{2} \) |
| 11 | \( 1 - 4.48T + 11T^{2} \) |
| 13 | \( 1 + 4.43T + 13T^{2} \) |
| 17 | \( 1 - 2.90T + 17T^{2} \) |
| 23 | \( 1 + 2.84T + 23T^{2} \) |
| 29 | \( 1 + 1.11T + 29T^{2} \) |
| 31 | \( 1 - 6.22T + 31T^{2} \) |
| 37 | \( 1 - 3.77T + 37T^{2} \) |
| 41 | \( 1 - 8.30T + 41T^{2} \) |
| 43 | \( 1 + 9.98T + 43T^{2} \) |
| 47 | \( 1 + 5.88T + 47T^{2} \) |
| 53 | \( 1 + 8.44T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 4.98T + 61T^{2} \) |
| 67 | \( 1 + 8.47T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 3.72T + 73T^{2} \) |
| 79 | \( 1 + 9.03T + 79T^{2} \) |
| 83 | \( 1 + 2.12T + 83T^{2} \) |
| 89 | \( 1 + 7.93T + 89T^{2} \) |
| 97 | \( 1 - 9.67T + 97T^{2} \) |
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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
−9.280348518815016541809849193491, −7.87489696097688176700052162512, −6.75438075247896739999333605444, −6.22903257339959368728716007293, −5.66047252117505559364157363350, −4.74859551964247702417842764991, −4.34752989850164000978516432661, −3.10298327831670570272976978284, −1.40657514029036047204288276562, 0,
1.40657514029036047204288276562, 3.10298327831670570272976978284, 4.34752989850164000978516432661, 4.74859551964247702417842764991, 5.66047252117505559364157363350, 6.22903257339959368728716007293, 6.75438075247896739999333605444, 7.87489696097688176700052162512, 9.280348518815016541809849193491
