| L(s) = 1 | − 4.42·2-s + 11.5·4-s − 31.6·7-s − 15.8·8-s + 11·11-s − 5.15·13-s + 140.·14-s − 22.6·16-s + 121.·17-s + 34.8·19-s − 48.6·22-s + 116.·23-s + 22.7·26-s − 366.·28-s + 69.4·29-s + 140.·31-s + 226.·32-s − 539.·34-s + 420.·37-s − 154.·38-s + 322.·41-s − 321.·43-s + 127.·44-s − 514.·46-s − 231.·47-s + 661.·49-s − 59.6·52-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 1.44·4-s − 1.71·7-s − 0.699·8-s + 0.301·11-s − 0.109·13-s + 2.67·14-s − 0.353·16-s + 1.73·17-s + 0.420·19-s − 0.471·22-s + 1.05·23-s + 0.171·26-s − 2.47·28-s + 0.444·29-s + 0.814·31-s + 1.25·32-s − 2.72·34-s + 1.86·37-s − 0.658·38-s + 1.22·41-s − 1.13·43-s + 0.436·44-s − 1.64·46-s − 0.718·47-s + 1.92·49-s − 0.159·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8042726409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8042726409\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 4.42T + 8T^{2} \) |
| 7 | \( 1 + 31.6T + 343T^{2} \) |
| 13 | \( 1 + 5.15T + 2.19e3T^{2} \) |
| 17 | \( 1 - 121.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 69.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 140.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 420.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 322.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 321.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 231.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 4.91T + 1.48e5T^{2} \) |
| 59 | \( 1 + 406.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 556.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 84.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 49.0T + 3.57e5T^{2} \) |
| 73 | \( 1 + 785.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 383.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 930.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 732.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.17e3T + 9.12e5T^{2} \) |
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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
−8.795759133585842790037926363210, −7.80968275803579131701313037068, −7.31943777769190295922530629347, −6.45017574975322343354199701980, −5.90904704424198216711065960583, −4.59061766979703980579529316556, −3.27799926686015587703525492087, −2.77023486848679734207366342686, −1.27496928258631777666068177414, −0.57126054679561116444234669376,
0.57126054679561116444234669376, 1.27496928258631777666068177414, 2.77023486848679734207366342686, 3.27799926686015587703525492087, 4.59061766979703980579529316556, 5.90904704424198216711065960583, 6.45017574975322343354199701980, 7.31943777769190295922530629347, 7.80968275803579131701313037068, 8.795759133585842790037926363210
