| L(s) = 1 | − 4·5-s − 7-s − 3·11-s + 4·13-s − 6·17-s + 8·19-s + 8·23-s + 11·25-s − 6·29-s − 2·31-s + 4·35-s + 5·37-s + 8·41-s − 9·43-s − 2·47-s + 49-s + 9·53-s + 12·55-s + 10·59-s − 10·61-s − 16·65-s − 3·67-s − 11·71-s − 16·73-s + 3·77-s + 5·79-s + 4·83-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.377·7-s − 0.904·11-s + 1.10·13-s − 1.45·17-s + 1.83·19-s + 1.66·23-s + 11/5·25-s − 1.11·29-s − 0.359·31-s + 0.676·35-s + 0.821·37-s + 1.24·41-s − 1.37·43-s − 0.291·47-s + 1/7·49-s + 1.23·53-s + 1.61·55-s + 1.30·59-s − 1.28·61-s − 1.98·65-s − 0.366·67-s − 1.30·71-s − 1.87·73-s + 0.341·77-s + 0.562·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.78648843816531542939913159606, −7.39663602717582615401955402317, −6.76106007903430555347453719104, −5.69657676529391470860991154544, −4.88927696151582815009946053484, −4.11188247989759469138679011635, −3.37517936578416847566575566664, −2.76097846926189978544698953609, −1.10531137529920236836655442589, 0,
1.10531137529920236836655442589, 2.76097846926189978544698953609, 3.37517936578416847566575566664, 4.11188247989759469138679011635, 4.88927696151582815009946053484, 5.69657676529391470860991154544, 6.76106007903430555347453719104, 7.39663602717582615401955402317, 7.78648843816531542939913159606
