| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 3.16·7-s + 8-s + 9-s − 2.16·11-s − 12-s − 1.16·13-s − 3.16·14-s + 16-s + 7.16·17-s + 18-s − 19-s + 3.16·21-s − 2.16·22-s + 7.32·23-s − 24-s − 1.16·26-s − 27-s − 3.16·28-s − 10.1·29-s − 7.32·31-s + 32-s + 2.16·33-s + 7.16·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.19·7-s + 0.353·8-s + 0.333·9-s − 0.651·11-s − 0.288·12-s − 0.322·13-s − 0.845·14-s + 0.250·16-s + 1.73·17-s + 0.235·18-s − 0.229·19-s + 0.690·21-s − 0.460·22-s + 1.52·23-s − 0.204·24-s − 0.227·26-s − 0.192·27-s − 0.597·28-s − 1.88·29-s − 1.31·31-s + 0.176·32-s + 0.376·33-s + 1.22·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.908181311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.908181311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 + 2.16T + 11T^{2} \) |
| 13 | \( 1 + 1.16T + 13T^{2} \) |
| 17 | \( 1 - 7.16T + 17T^{2} \) |
| 23 | \( 1 - 7.32T + 23T^{2} \) |
| 29 | \( 1 + 10.1T + 29T^{2} \) |
| 31 | \( 1 + 7.32T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 6.32T + 41T^{2} \) |
| 43 | \( 1 + 2.83T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 8.16T + 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 - 2.16T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 5.16T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 - 7.32T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 5.32T + 89T^{2} \) |
| 97 | \( 1 - 19.4T + 97T^{2} \) |
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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
−8.923105543831959452592431231343, −7.51780013003489545750272305722, −7.35326246101385024710560693505, −6.29598148765238379900483252366, −5.59844681752436845088987230563, −5.16120985443499075813917329518, −3.93227443534041960253736021553, −3.28672443233261357972023514643, −2.32568856046296764835560675109, −0.77644326464205890941869909154,
0.77644326464205890941869909154, 2.32568856046296764835560675109, 3.28672443233261357972023514643, 3.93227443534041960253736021553, 5.16120985443499075813917329518, 5.59844681752436845088987230563, 6.29598148765238379900483252366, 7.35326246101385024710560693505, 7.51780013003489545750272305722, 8.923105543831959452592431231343
