| L(s) = 1 | + 2.61·2-s − 3-s + 4.81·4-s + 3.81·5-s − 2.61·6-s + 7.34·8-s + 9-s + 9.95·10-s − 4.73·11-s − 4.81·12-s − 13-s − 3.81·15-s + 9.55·16-s + 5.22·17-s + 2.61·18-s − 2.92·19-s + 18.3·20-s − 12.3·22-s + 3.33·23-s − 7.34·24-s + 9.55·25-s − 2.61·26-s − 27-s − 0.922·29-s − 9.95·30-s + 7.51·31-s + 10.2·32-s + ⋯ |
| L(s) = 1 | + 1.84·2-s − 0.577·3-s + 2.40·4-s + 1.70·5-s − 1.06·6-s + 2.59·8-s + 0.333·9-s + 3.14·10-s − 1.42·11-s − 1.38·12-s − 0.277·13-s − 0.984·15-s + 2.38·16-s + 1.26·17-s + 0.615·18-s − 0.670·19-s + 4.10·20-s − 2.63·22-s + 0.694·23-s − 1.49·24-s + 1.91·25-s − 0.511·26-s − 0.192·27-s − 0.171·29-s − 1.81·30-s + 1.35·31-s + 1.81·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.950450659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.950450659\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 5 | \( 1 - 3.81T + 5T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 + 2.92T + 19T^{2} \) |
| 23 | \( 1 - 3.33T + 23T^{2} \) |
| 29 | \( 1 + 0.922T + 29T^{2} \) |
| 31 | \( 1 - 7.51T + 31T^{2} \) |
| 37 | \( 1 - 0.154T + 37T^{2} \) |
| 41 | \( 1 + 6.36T + 41T^{2} \) |
| 43 | \( 1 + 6.55T + 43T^{2} \) |
| 47 | \( 1 + 9.03T + 47T^{2} \) |
| 53 | \( 1 - 8.55T + 53T^{2} \) |
| 59 | \( 1 + 3.95T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 6.58T + 71T^{2} \) |
| 73 | \( 1 - 7.73T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 1.40T + 83T^{2} \) |
| 89 | \( 1 - 1.96T + 89T^{2} \) |
| 97 | \( 1 - 2.11T + 97T^{2} \) |
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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
−9.572092235490181032075627935793, −8.161111632574951171463277947516, −7.15843899322385305235108967649, −6.34868069096053002103989244422, −5.83559009690674376137725567846, −5.08642110563666370220060422999, −4.79564159119713193983318154562, −3.25830501722698521304745116018, −2.53705785222885507365364174128, −1.57733661967134253562589763215,
1.57733661967134253562589763215, 2.53705785222885507365364174128, 3.25830501722698521304745116018, 4.79564159119713193983318154562, 5.08642110563666370220060422999, 5.83559009690674376137725567846, 6.34868069096053002103989244422, 7.15843899322385305235108967649, 8.161111632574951171463277947516, 9.572092235490181032075627935793
