| L(s) = 1 | + 5-s − 5·7-s − 2·11-s − 13-s + 3·17-s + 2·19-s + 4·23-s − 4·25-s + 6·29-s + 4·31-s − 5·35-s + 11·37-s − 8·41-s + 43-s + 9·47-s + 18·49-s + 12·53-s − 2·55-s + 6·59-s − 65-s − 6·67-s + 7·71-s − 2·73-s + 10·77-s − 12·79-s − 16·83-s + 3·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.88·7-s − 0.603·11-s − 0.277·13-s + 0.727·17-s + 0.458·19-s + 0.834·23-s − 4/5·25-s + 1.11·29-s + 0.718·31-s − 0.845·35-s + 1.80·37-s − 1.24·41-s + 0.152·43-s + 1.31·47-s + 18/7·49-s + 1.64·53-s − 0.269·55-s + 0.781·59-s − 0.124·65-s − 0.733·67-s + 0.830·71-s − 0.234·73-s + 1.13·77-s − 1.35·79-s − 1.75·83-s + 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.366716428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.366716428\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−9.399377966647754514275339054317, −8.571559742557125580918706419247, −7.51880508752900332556843831011, −6.82508995489029179736833518542, −6.01472830731285355227624329326, −5.41568943920328063726692378316, −4.19080409729406201912090076952, −3.11607344408057507029997990678, −2.55679172121781849822247168687, −0.77347272515720828453769161406,
0.77347272515720828453769161406, 2.55679172121781849822247168687, 3.11607344408057507029997990678, 4.19080409729406201912090076952, 5.41568943920328063726692378316, 6.01472830731285355227624329326, 6.82508995489029179736833518542, 7.51880508752900332556843831011, 8.571559742557125580918706419247, 9.399377966647754514275339054317
