L(s) = 1 | − 3-s + 0.356·5-s + 4.08·7-s + 9-s + 0.568·11-s − 3.27·13-s − 0.356·15-s − 0.916·17-s + 5.34·19-s − 4.08·21-s − 3.28·23-s − 4.87·25-s − 27-s − 5.53·29-s + 0.909·31-s − 0.568·33-s + 1.45·35-s − 8.81·37-s + 3.27·39-s − 3.51·41-s − 10.3·43-s + 0.356·45-s + 4.98·47-s + 9.71·49-s + 0.916·51-s − 9.48·53-s + 0.202·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.159·5-s + 1.54·7-s + 0.333·9-s + 0.171·11-s − 0.909·13-s − 0.0920·15-s − 0.222·17-s + 1.22·19-s − 0.892·21-s − 0.684·23-s − 0.974·25-s − 0.192·27-s − 1.02·29-s + 0.163·31-s − 0.0988·33-s + 0.246·35-s − 1.44·37-s + 0.524·39-s − 0.548·41-s − 1.57·43-s + 0.0531·45-s + 0.726·47-s + 1.38·49-s + 0.128·51-s − 1.30·53-s + 0.0273·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 251 | \( 1 - T \) |
good | 5 | \( 1 - 0.356T + 5T^{2} \) |
| 7 | \( 1 - 4.08T + 7T^{2} \) |
| 11 | \( 1 - 0.568T + 11T^{2} \) |
| 13 | \( 1 + 3.27T + 13T^{2} \) |
| 17 | \( 1 + 0.916T + 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 + 3.28T + 23T^{2} \) |
| 29 | \( 1 + 5.53T + 29T^{2} \) |
| 31 | \( 1 - 0.909T + 31T^{2} \) |
| 37 | \( 1 + 8.81T + 37T^{2} \) |
| 41 | \( 1 + 3.51T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 4.98T + 47T^{2} \) |
| 53 | \( 1 + 9.48T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 7.13T + 67T^{2} \) |
| 71 | \( 1 + 5.06T + 71T^{2} \) |
| 73 | \( 1 - 0.192T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 7.60T + 83T^{2} \) |
| 89 | \( 1 + 7.97T + 89T^{2} \) |
| 97 | \( 1 - 18.9T + 97T^{2} \) |
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
−7.55756555871154592478603182719, −7.24700947817162392326538412077, −6.18748720523356213077863080030, −5.41357828140996981747672180007, −4.97355341747187973596839542395, −4.26289538376737754211652156986, −3.28153472519712839067397232684, −1.99366150022901757112574925673, −1.49231544435733245107828086310, 0,
1.49231544435733245107828086310, 1.99366150022901757112574925673, 3.28153472519712839067397232684, 4.26289538376737754211652156986, 4.97355341747187973596839542395, 5.41357828140996981747672180007, 6.18748720523356213077863080030, 7.24700947817162392326538412077, 7.55756555871154592478603182719