| L(s) = 1 | − 3-s − 2.90·5-s + 2.88·7-s + 9-s + 3.21·11-s + 2.90·15-s + 5.12·17-s + 1.45·19-s − 2.88·21-s − 4.09·23-s + 3.42·25-s − 27-s + 3.45·29-s − 7.55·31-s − 3.21·33-s − 8.36·35-s + 11.4·37-s − 7.52·41-s + 6.51·43-s − 2.90·45-s + 9.09·47-s + 1.30·49-s − 5.12·51-s + 5.65·53-s − 9.31·55-s − 1.45·57-s − 7.21·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.29·5-s + 1.08·7-s + 0.333·9-s + 0.968·11-s + 0.749·15-s + 1.24·17-s + 0.333·19-s − 0.628·21-s − 0.853·23-s + 0.684·25-s − 0.192·27-s + 0.641·29-s − 1.35·31-s − 0.559·33-s − 1.41·35-s + 1.87·37-s − 1.17·41-s + 0.994·43-s − 0.432·45-s + 1.32·47-s + 0.186·49-s − 0.718·51-s + 0.776·53-s − 1.25·55-s − 0.192·57-s − 0.939·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.461844180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.461844180\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2.90T + 5T^{2} \) |
| 7 | \( 1 - 2.88T + 7T^{2} \) |
| 11 | \( 1 - 3.21T + 11T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 - 1.45T + 19T^{2} \) |
| 23 | \( 1 + 4.09T + 23T^{2} \) |
| 29 | \( 1 - 3.45T + 29T^{2} \) |
| 31 | \( 1 + 7.55T + 31T^{2} \) |
| 37 | \( 1 - 11.4T + 37T^{2} \) |
| 41 | \( 1 + 7.52T + 41T^{2} \) |
| 43 | \( 1 - 6.51T + 43T^{2} \) |
| 47 | \( 1 - 9.09T + 47T^{2} \) |
| 53 | \( 1 - 5.65T + 53T^{2} \) |
| 59 | \( 1 + 7.21T + 59T^{2} \) |
| 61 | \( 1 + 1.55T + 61T^{2} \) |
| 67 | \( 1 - 5.43T + 67T^{2} \) |
| 71 | \( 1 + 15.2T + 71T^{2} \) |
| 73 | \( 1 + 15.4T + 73T^{2} \) |
| 79 | \( 1 - 5.94T + 79T^{2} \) |
| 83 | \( 1 + 8.20T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−8.290170465528163781889782861823, −7.56284699259670125841315671261, −7.29763862295626428591927767277, −6.11113469259349442071875110798, −5.49466765882609172549812915266, −4.45798548760392880113716311605, −4.09690725026686363425458980751, −3.15729618099727307254905857397, −1.69491092925796075509250822729, −0.75596601709637103429745586657,
0.75596601709637103429745586657, 1.69491092925796075509250822729, 3.15729618099727307254905857397, 4.09690725026686363425458980751, 4.45798548760392880113716311605, 5.49466765882609172549812915266, 6.11113469259349442071875110798, 7.29763862295626428591927767277, 7.56284699259670125841315671261, 8.290170465528163781889782861823
