| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 12-s + 13-s − 14-s + 16-s + 6·17-s + 18-s − 5·19-s − 21-s − 23-s + 24-s − 5·25-s + 26-s + 27-s − 28-s − 6·29-s − 31-s + 32-s + 6·34-s + 36-s + 11·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.277·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.14·19-s − 0.218·21-s − 0.208·23-s + 0.204·24-s − 25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.179·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 1.80·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.257970624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.257970624\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.50358709839461, −13.13632989763743, −12.79565807248802, −12.26306578478846, −11.72544644671418, −11.27595391828094, −10.72591037968694, −10.09716555774909, −9.712028029690627, −9.358846773411018, −8.436654852333543, −8.193245980624859, −7.661448726308886, −7.057791468196405, −6.567471462663057, −5.980691652864508, −5.494396745457558, −5.033577377036722, −4.070636604202205, −3.819096937770545, −3.454980330703829, −2.426235699668501, −2.306868575929389, −1.382468437779346, −0.5965586795699195,
0.5965586795699195, 1.382468437779346, 2.306868575929389, 2.426235699668501, 3.454980330703829, 3.819096937770545, 4.070636604202205, 5.033577377036722, 5.494396745457558, 5.980691652864508, 6.567471462663057, 7.057791468196405, 7.661448726308886, 8.193245980624859, 8.436654852333543, 9.358846773411018, 9.712028029690627, 10.09716555774909, 10.72591037968694, 11.27595391828094, 11.72544644671418, 12.26306578478846, 12.79565807248802, 13.13632989763743, 13.50358709839461
