L(s) = 1 | + 5-s − 3·7-s + 4.31·11-s − 2.31·13-s + 3.31·17-s − 1.68·19-s − 23-s + 25-s + 0.683·29-s − 9.63·31-s − 3·35-s + 11.6·37-s − 3.31·41-s + 2.63·43-s − 10.9·47-s + 2·49-s + 9.94·53-s + 4.31·55-s + 11.9·59-s + 3.68·61-s − 2.31·65-s + 2.36·67-s + 7.94·71-s + 1.68·73-s − 12.9·77-s − 5.31·83-s + 3.31·85-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.13·7-s + 1.30·11-s − 0.642·13-s + 0.804·17-s − 0.386·19-s − 0.208·23-s + 0.200·25-s + 0.126·29-s − 1.73·31-s − 0.507·35-s + 1.91·37-s − 0.517·41-s + 0.401·43-s − 1.59·47-s + 0.285·49-s + 1.36·53-s + 0.582·55-s + 1.55·59-s + 0.471·61-s − 0.287·65-s + 0.289·67-s + 0.943·71-s + 0.197·73-s − 1.47·77-s − 0.583·83-s + 0.359·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.882842590\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.882842590\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 13 | \( 1 + 2.31T + 13T^{2} \) |
| 17 | \( 1 - 3.31T + 17T^{2} \) |
| 19 | \( 1 + 1.68T + 19T^{2} \) |
| 29 | \( 1 - 0.683T + 29T^{2} \) |
| 31 | \( 1 + 9.63T + 31T^{2} \) |
| 37 | \( 1 - 11.6T + 37T^{2} \) |
| 41 | \( 1 + 3.31T + 41T^{2} \) |
| 43 | \( 1 - 2.63T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 9.94T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 3.68T + 61T^{2} \) |
| 67 | \( 1 - 2.36T + 67T^{2} \) |
| 71 | \( 1 - 7.94T + 71T^{2} \) |
| 73 | \( 1 - 1.68T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 5.31T + 83T^{2} \) |
| 89 | \( 1 + 8.63T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−7.70983956786685106767244335223, −6.95766462884354972580795919740, −6.47622105898560459342202031306, −5.83298923175957697722262128638, −5.13531030668431488674381214115, −4.08316739956221152938840292725, −3.56441816367635607915807645338, −2.68224845031852740083856478415, −1.78540791522464592475425805967, −0.67399249946092103126652483353,
0.67399249946092103126652483353, 1.78540791522464592475425805967, 2.68224845031852740083856478415, 3.56441816367635607915807645338, 4.08316739956221152938840292725, 5.13531030668431488674381214115, 5.83298923175957697722262128638, 6.47622105898560459342202031306, 6.95766462884354972580795919740, 7.70983956786685106767244335223