| L(s) = 1 | + 3-s − 4·5-s + 4·7-s − 2·9-s + 5·11-s − 4·13-s − 4·15-s − 2·17-s + 4·21-s + 11·25-s − 5·27-s + 4·29-s + 8·31-s + 5·33-s − 16·35-s + 8·37-s − 4·39-s + 3·41-s + 8·43-s + 8·45-s + 12·47-s + 9·49-s − 2·51-s + 12·53-s − 20·55-s + 3·59-s + 4·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1.51·7-s − 2/3·9-s + 1.50·11-s − 1.10·13-s − 1.03·15-s − 0.485·17-s + 0.872·21-s + 11/5·25-s − 0.962·27-s + 0.742·29-s + 1.43·31-s + 0.870·33-s − 2.70·35-s + 1.31·37-s − 0.640·39-s + 0.468·41-s + 1.21·43-s + 1.19·45-s + 1.75·47-s + 9/7·49-s − 0.280·51-s + 1.64·53-s − 2.69·55-s + 0.390·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.268475564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.268475564\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−14.00760048918921, −13.61294602913741, −12.59544805490914, −12.17136855880336, −11.83575077197303, −11.45993123486353, −11.12226970656409, −10.56017258853903, −9.781443940316640, −9.031020604952139, −8.825638368892465, −8.202009852082620, −7.923477651441484, −7.420403856702690, −6.960994684371047, −6.284479941578751, −5.507175149169247, −4.778292196499804, −4.328566671590741, −4.070182918727547, −3.398132620566323, −2.488993248340802, −2.292361155678157, −0.9847267795963869, −0.7096237044937162,
0.7096237044937162, 0.9847267795963869, 2.292361155678157, 2.488993248340802, 3.398132620566323, 4.070182918727547, 4.328566671590741, 4.778292196499804, 5.507175149169247, 6.284479941578751, 6.960994684371047, 7.420403856702690, 7.923477651441484, 8.202009852082620, 8.825638368892465, 9.031020604952139, 9.781443940316640, 10.56017258853903, 11.12226970656409, 11.45993123486353, 11.83575077197303, 12.17136855880336, 12.59544805490914, 13.61294602913741, 14.00760048918921
