| L(s) = 1 | − 5-s + 3.37·7-s − 1.37·11-s + 13-s − 7.37·17-s − 4·19-s − 7.37·23-s + 25-s + 2.74·29-s − 6.74·31-s − 3.37·35-s − 8.11·37-s + 1.37·41-s + 10.7·43-s + 11.4·47-s + 4.37·49-s − 7.37·53-s + 1.37·55-s − 8.74·59-s + 6.62·61-s − 65-s − 4·67-s − 1.37·71-s + 7.48·73-s − 4.62·77-s − 16.8·79-s − 3.25·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.27·7-s − 0.413·11-s + 0.277·13-s − 1.78·17-s − 0.917·19-s − 1.53·23-s + 0.200·25-s + 0.509·29-s − 1.21·31-s − 0.570·35-s − 1.33·37-s + 0.214·41-s + 1.63·43-s + 1.67·47-s + 0.624·49-s − 1.01·53-s + 0.185·55-s − 1.13·59-s + 0.848·61-s − 0.124·65-s − 0.488·67-s − 0.162·71-s + 0.876·73-s − 0.527·77-s − 1.89·79-s − 0.357·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 17 | \( 1 + 7.37T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 7.37T + 23T^{2} \) |
| 29 | \( 1 - 2.74T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 + 8.11T + 37T^{2} \) |
| 41 | \( 1 - 1.37T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 7.37T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 - 6.62T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 1.37T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 + 3.25T + 83T^{2} \) |
| 89 | \( 1 - 4.11T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.582041332167730929035646662541, −7.905310436139609331133831169540, −7.19845619347951974394342910805, −6.26488849128986717078509648254, −5.38642467498859461681626274934, −4.41350125105966121611828288026, −4.01366997485287848565259445128, −2.49946825346163621314334396795, −1.70880617446467909880194200828, 0,
1.70880617446467909880194200828, 2.49946825346163621314334396795, 4.01366997485287848565259445128, 4.41350125105966121611828288026, 5.38642467498859461681626274934, 6.26488849128986717078509648254, 7.19845619347951974394342910805, 7.905310436139609331133831169540, 8.582041332167730929035646662541
