| L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 7-s + 8-s + 9-s + 2·10-s + 12-s + 4.47·13-s + 14-s + 2·15-s + 16-s − 4.47·17-s + 18-s − 6.47·19-s + 2·20-s + 21-s − 23-s + 24-s − 25-s + 4.47·26-s + 27-s + 28-s − 2·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.894·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.632·10-s + 0.288·12-s + 1.24·13-s + 0.267·14-s + 0.516·15-s + 0.250·16-s − 1.08·17-s + 0.235·18-s − 1.48·19-s + 0.447·20-s + 0.218·21-s − 0.208·23-s + 0.204·24-s − 0.200·25-s + 0.877·26-s + 0.192·27-s + 0.188·28-s − 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.554775313\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.554775313\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 - 6.94T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 + 6.47T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 - 8.47T + 97T^{2} \) |
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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
−10.26124258674917254550487217205, −8.906248210020816594514505143451, −8.605673790472952672294072178775, −7.37588311853937904325025269121, −6.38385247650647585026092836777, −5.81434742854424470904257199085, −4.58425400030105124017947836583, −3.82024851165530888117705794237, −2.49878402826380940678274663916, −1.67695778472700422371539025888,
1.67695778472700422371539025888, 2.49878402826380940678274663916, 3.82024851165530888117705794237, 4.58425400030105124017947836583, 5.81434742854424470904257199085, 6.38385247650647585026092836777, 7.37588311853937904325025269121, 8.605673790472952672294072178775, 8.906248210020816594514505143451, 10.26124258674917254550487217205
