L(s) = 1 | + 4·5-s + 7-s − 2·11-s − 4·13-s − 4·17-s + 19-s + 6·23-s + 11·25-s + 2·29-s + 4·31-s + 4·35-s + 2·37-s + 2·41-s − 8·43-s − 4·47-s + 49-s + 10·53-s − 8·55-s + 4·59-s + 6·61-s − 16·65-s − 10·67-s + 16·71-s − 6·73-s − 2·77-s − 10·79-s + 12·83-s + ⋯ |
L(s) = 1 | + 1.78·5-s + 0.377·7-s − 0.603·11-s − 1.10·13-s − 0.970·17-s + 0.229·19-s + 1.25·23-s + 11/5·25-s + 0.371·29-s + 0.718·31-s + 0.676·35-s + 0.328·37-s + 0.312·41-s − 1.21·43-s − 0.583·47-s + 1/7·49-s + 1.37·53-s − 1.07·55-s + 0.520·59-s + 0.768·61-s − 1.98·65-s − 1.22·67-s + 1.89·71-s − 0.702·73-s − 0.227·77-s − 1.12·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.010510186\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.010510186\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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
−7.56780738802863319034520298588, −6.89752996430020665129881005658, −6.35803456865051606172706780969, −5.53563308511445108037596263224, −5.00800973880088911718029672915, −4.54976813297965541155846494959, −3.13184770993157943602511372045, −2.43858276717934527568061755688, −1.93193903687707711736862198813, −0.826832037639397092131755132838,
0.826832037639397092131755132838, 1.93193903687707711736862198813, 2.43858276717934527568061755688, 3.13184770993157943602511372045, 4.54976813297965541155846494959, 5.00800973880088911718029672915, 5.53563308511445108037596263224, 6.35803456865051606172706780969, 6.89752996430020665129881005658, 7.56780738802863319034520298588