| L(s) = 1 | − 5.20·3-s − 5·5-s + 15.1·7-s + 0.117·9-s − 3.79·11-s + 21.8·13-s + 26.0·15-s + 5.42·17-s − 129.·19-s − 79.1·21-s + 58.3·23-s + 25·25-s + 139.·27-s + 29·29-s − 36.2·31-s + 19.7·33-s − 75.9·35-s + 182.·37-s − 113.·39-s − 189.·41-s + 238.·43-s − 0.586·45-s + 94.4·47-s − 112.·49-s − 28.2·51-s + 123.·53-s + 18.9·55-s + ⋯ |
| L(s) = 1 | − 1.00·3-s − 0.447·5-s + 0.820·7-s + 0.00434·9-s − 0.104·11-s + 0.466·13-s + 0.448·15-s + 0.0773·17-s − 1.56·19-s − 0.822·21-s + 0.528·23-s + 0.200·25-s + 0.997·27-s + 0.185·29-s − 0.209·31-s + 0.104·33-s − 0.366·35-s + 0.811·37-s − 0.467·39-s − 0.720·41-s + 0.846·43-s − 0.00194·45-s + 0.292·47-s − 0.327·49-s − 0.0775·51-s + 0.319·53-s + 0.0465·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 5.20T + 27T^{2} \) |
| 7 | \( 1 - 15.1T + 343T^{2} \) |
| 11 | \( 1 + 3.79T + 1.33e3T^{2} \) |
| 13 | \( 1 - 21.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 5.42T + 4.91e3T^{2} \) |
| 19 | \( 1 + 129.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 58.3T + 1.21e4T^{2} \) |
| 31 | \( 1 + 36.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 182.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 189.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 238.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 94.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 123.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 836.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 667.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 657.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 313.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 573.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 408.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 273.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.44e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 73.6T + 9.12e5T^{2} \) |
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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
−8.740592605014439583000210438236, −8.321273210544961643182324042374, −7.25263369626383279386729731299, −6.39637843638587056975740482690, −5.58720537380202361384213691791, −4.75303622966634877976289811765, −3.94586692933649404916740673098, −2.52816332916665470416455967233, −1.16186218263449860805157123776, 0,
1.16186218263449860805157123776, 2.52816332916665470416455967233, 3.94586692933649404916740673098, 4.75303622966634877976289811765, 5.58720537380202361384213691791, 6.39637843638587056975740482690, 7.25263369626383279386729731299, 8.321273210544961643182324042374, 8.740592605014439583000210438236
