| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 3·7-s − 8-s + 9-s + 5.77·11-s + 12-s − 3.77·13-s + 3·14-s + 16-s + 0.772·17-s − 18-s + 7.77·19-s − 3·21-s − 5.77·22-s + 23-s − 24-s + 3.77·26-s + 27-s − 3·28-s + 3·29-s − 9.54·31-s − 32-s + 5.77·33-s − 0.772·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 0.333·9-s + 1.74·11-s + 0.288·12-s − 1.04·13-s + 0.801·14-s + 0.250·16-s + 0.187·17-s − 0.235·18-s + 1.78·19-s − 0.654·21-s − 1.23·22-s + 0.208·23-s − 0.204·24-s + 0.739·26-s + 0.192·27-s − 0.566·28-s + 0.557·29-s − 1.71·31-s − 0.176·32-s + 1.00·33-s − 0.132·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.573499019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.573499019\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 5.77T + 11T^{2} \) |
| 13 | \( 1 + 3.77T + 13T^{2} \) |
| 17 | \( 1 - 0.772T + 17T^{2} \) |
| 19 | \( 1 - 7.77T + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 9.54T + 31T^{2} \) |
| 37 | \( 1 - 6.77T + 37T^{2} \) |
| 41 | \( 1 + 5.77T + 41T^{2} \) |
| 43 | \( 1 - 7.77T + 43T^{2} \) |
| 47 | \( 1 + 8.77T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 + 9.54T + 61T^{2} \) |
| 67 | \( 1 + 9.54T + 67T^{2} \) |
| 71 | \( 1 + 4.77T + 71T^{2} \) |
| 73 | \( 1 - 6.54T + 73T^{2} \) |
| 79 | \( 1 - 2.22T + 79T^{2} \) |
| 83 | \( 1 + T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−8.952548758755425993860558627203, −7.72532875423666034563636185527, −7.25898045343318467564135063167, −6.58627516889358795843683636662, −5.82512385636807742735454555055, −4.69059877759759686647558142392, −3.53930909040159474666659532958, −3.11572604966901884782002356794, −1.92510402792346662634850703788, −0.815943988702373896305767237976,
0.815943988702373896305767237976, 1.92510402792346662634850703788, 3.11572604966901884782002356794, 3.53930909040159474666659532958, 4.69059877759759686647558142392, 5.82512385636807742735454555055, 6.58627516889358795843683636662, 7.25898045343318467564135063167, 7.72532875423666034563636185527, 8.952548758755425993860558627203
