| L(s) = 1 | + 3.18·5-s − 4.42·7-s − 11-s − 13-s + 1.49·17-s − 1.76·19-s − 1.61·23-s + 5.17·25-s + 9.36·29-s − 5.31·31-s − 14.1·35-s − 0.810·37-s + 1.91·41-s + 5.23·43-s − 8.37·47-s + 12.6·49-s − 6.61·53-s − 3.18·55-s − 11.4·59-s + 12.2·61-s − 3.18·65-s − 12.0·67-s + 10.4·71-s − 13.8·73-s + 4.42·77-s − 2.34·79-s + 0.935·83-s + ⋯ |
| L(s) = 1 | + 1.42·5-s − 1.67·7-s − 0.301·11-s − 0.277·13-s + 0.361·17-s − 0.404·19-s − 0.337·23-s + 1.03·25-s + 1.73·29-s − 0.955·31-s − 2.38·35-s − 0.133·37-s + 0.299·41-s + 0.798·43-s − 1.22·47-s + 1.80·49-s − 0.908·53-s − 0.430·55-s − 1.48·59-s + 1.56·61-s − 0.395·65-s − 1.47·67-s + 1.23·71-s − 1.62·73-s + 0.504·77-s − 0.263·79-s + 0.102·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 3.18T + 5T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 17 | \( 1 - 1.49T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 + 1.61T + 23T^{2} \) |
| 29 | \( 1 - 9.36T + 29T^{2} \) |
| 31 | \( 1 + 5.31T + 31T^{2} \) |
| 37 | \( 1 + 0.810T + 37T^{2} \) |
| 41 | \( 1 - 1.91T + 41T^{2} \) |
| 43 | \( 1 - 5.23T + 43T^{2} \) |
| 47 | \( 1 + 8.37T + 47T^{2} \) |
| 53 | \( 1 + 6.61T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 12.0T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 13.8T + 73T^{2} \) |
| 79 | \( 1 + 2.34T + 79T^{2} \) |
| 83 | \( 1 - 0.935T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 3.18T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.83227697300072008694641925519, −6.86798734625485728541574265595, −6.37915190064425056202874576621, −5.83698911140593496909743789232, −5.12368448411121261840880847035, −4.08105320789613843914187637942, −3.02757612353822848486876993660, −2.55465279493661080206024508624, −1.43666811454205369359952525452, 0,
1.43666811454205369359952525452, 2.55465279493661080206024508624, 3.02757612353822848486876993660, 4.08105320789613843914187637942, 5.12368448411121261840880847035, 5.83698911140593496909743789232, 6.37915190064425056202874576621, 6.86798734625485728541574265595, 7.83227697300072008694641925519
