| L(s) = 1 | + 1.34·2-s + 2.02·3-s − 0.202·4-s + 2.71·6-s + 7-s − 2.95·8-s + 1.11·9-s − 4.63·11-s − 0.411·12-s − 1.05·13-s + 1.34·14-s − 3.55·16-s + 0.442·17-s + 1.49·18-s − 0.296·19-s + 2.02·21-s − 6.21·22-s + 23-s − 5.98·24-s − 1.40·26-s − 3.82·27-s − 0.202·28-s − 6.48·29-s − 2.99·31-s + 1.14·32-s − 9.39·33-s + 0.593·34-s + ⋯ |
| L(s) = 1 | + 0.947·2-s + 1.17·3-s − 0.101·4-s + 1.10·6-s + 0.377·7-s − 1.04·8-s + 0.370·9-s − 1.39·11-s − 0.118·12-s − 0.291·13-s + 0.358·14-s − 0.888·16-s + 0.107·17-s + 0.351·18-s − 0.0679·19-s + 0.442·21-s − 1.32·22-s + 0.208·23-s − 1.22·24-s − 0.276·26-s − 0.736·27-s − 0.0383·28-s − 1.20·29-s − 0.538·31-s + 0.202·32-s − 1.63·33-s + 0.101·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.34T + 2T^{2} \) |
| 3 | \( 1 - 2.02T + 3T^{2} \) |
| 11 | \( 1 + 4.63T + 11T^{2} \) |
| 13 | \( 1 + 1.05T + 13T^{2} \) |
| 17 | \( 1 - 0.442T + 17T^{2} \) |
| 19 | \( 1 + 0.296T + 19T^{2} \) |
| 29 | \( 1 + 6.48T + 29T^{2} \) |
| 31 | \( 1 + 2.99T + 31T^{2} \) |
| 37 | \( 1 - 0.971T + 37T^{2} \) |
| 41 | \( 1 + 6.58T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 3.92T + 47T^{2} \) |
| 53 | \( 1 - 0.592T + 53T^{2} \) |
| 59 | \( 1 - 6.10T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 0.569T + 71T^{2} \) |
| 73 | \( 1 - 9.38T + 73T^{2} \) |
| 79 | \( 1 + 1.57T + 79T^{2} \) |
| 83 | \( 1 - 5.03T + 83T^{2} \) |
| 89 | \( 1 + 3.61T + 89T^{2} \) |
| 97 | \( 1 - 3.01T + 97T^{2} \) |
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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
−8.103822698058666517197488920338, −7.52783314518418886728142513339, −6.56637890619001070618362814492, −5.36988636507896928238094639876, −5.22197084817437745702110221125, −4.13111621908596382149712192644, −3.40947883128762172890702470771, −2.73474623944536536469891814455, −1.93791232230168940390655716939, 0,
1.93791232230168940390655716939, 2.73474623944536536469891814455, 3.40947883128762172890702470771, 4.13111621908596382149712192644, 5.22197084817437745702110221125, 5.36988636507896928238094639876, 6.56637890619001070618362814492, 7.52783314518418886728142513339, 8.103822698058666517197488920338
