L(s) = 1 | + 7-s − 3·9-s − 4·11-s − 3·13-s − 17-s + 23-s + 4·31-s + 11·37-s − 10·41-s + 2·43-s − 11·47-s + 49-s + 53-s + 8·59-s − 8·61-s − 3·63-s + 4·71-s − 4·73-s − 4·77-s + 11·79-s + 9·81-s + 13·83-s + 89-s − 3·91-s − 7·97-s + 12·99-s + 101-s + ⋯ |
L(s) = 1 | + 0.377·7-s − 9-s − 1.20·11-s − 0.832·13-s − 0.242·17-s + 0.208·23-s + 0.718·31-s + 1.80·37-s − 1.56·41-s + 0.304·43-s − 1.60·47-s + 1/7·49-s + 0.137·53-s + 1.04·59-s − 1.02·61-s − 0.377·63-s + 0.474·71-s − 0.468·73-s − 0.455·77-s + 1.23·79-s + 81-s + 1.42·83-s + 0.105·89-s − 0.314·91-s − 0.710·97-s + 1.20·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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
−14.60901069474202, −13.90871605053317, −13.51693781025779, −13.02691920303076, −12.49244131290553, −11.86161932189599, −11.48839908333899, −10.99249681001692, −10.44478964133731, −9.923782461360197, −9.442514217180024, −8.737053966521865, −8.261816321230843, −7.815443571298131, −7.398407806308400, −6.524192195060701, −6.171028152273942, −5.321167073805485, −5.034180964047580, −4.536197247899587, −3.629732723612799, −2.918510462122134, −2.509995773446112, −1.868088003092780, −0.7676129791313198, 0,
0.7676129791313198, 1.868088003092780, 2.509995773446112, 2.918510462122134, 3.629732723612799, 4.536197247899587, 5.034180964047580, 5.321167073805485, 6.171028152273942, 6.524192195060701, 7.398407806308400, 7.815443571298131, 8.261816321230843, 8.737053966521865, 9.442514217180024, 9.923782461360197, 10.44478964133731, 10.99249681001692, 11.48839908333899, 11.86161932189599, 12.49244131290553, 13.02691920303076, 13.51693781025779, 13.90871605053317, 14.60901069474202