L(s) = 1 | − 3-s + 2·5-s + 3·7-s − 2·9-s + 3·11-s + 6·13-s − 2·15-s + 5·17-s + 2·19-s − 3·21-s + 4·23-s − 25-s + 5·27-s + 7·29-s − 5·31-s − 3·33-s + 6·35-s + 11·37-s − 6·39-s − 2·41-s − 8·43-s − 4·45-s + 2·49-s − 5·51-s − 6·53-s + 6·55-s − 2·57-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.13·7-s − 2/3·9-s + 0.904·11-s + 1.66·13-s − 0.516·15-s + 1.21·17-s + 0.458·19-s − 0.654·21-s + 0.834·23-s − 1/5·25-s + 0.962·27-s + 1.29·29-s − 0.898·31-s − 0.522·33-s + 1.01·35-s + 1.80·37-s − 0.960·39-s − 0.312·41-s − 1.21·43-s − 0.596·45-s + 2/7·49-s − 0.700·51-s − 0.824·53-s + 0.809·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.767848310\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.767848310\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 83 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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
−8.329893308334208368803964563128, −7.50704744345289358390695990950, −6.43849364286368598220680598083, −6.03908971095412582423394529075, −5.39435567060411867274578529850, −4.73425169524415203719187504641, −3.69889287656033475652720645024, −2.84692566072845257253775167246, −1.51392803132280425667443874400, −1.10159956738760523747148561916,
1.10159956738760523747148561916, 1.51392803132280425667443874400, 2.84692566072845257253775167246, 3.69889287656033475652720645024, 4.73425169524415203719187504641, 5.39435567060411867274578529850, 6.03908971095412582423394529075, 6.43849364286368598220680598083, 7.50704744345289358390695990950, 8.329893308334208368803964563128