| L(s) = 1 | − 0.0533·2-s − 1.99·4-s + 0.751·5-s + 2.49·7-s + 0.213·8-s − 0.0400·10-s + 5.03·11-s − 1.59·13-s − 0.133·14-s + 3.98·16-s − 7.25·17-s − 4.33·19-s − 1.50·20-s − 0.268·22-s − 3.74·23-s − 4.43·25-s + 0.0851·26-s − 4.98·28-s + 2.31·29-s − 2.25·31-s − 0.639·32-s + 0.387·34-s + 1.87·35-s − 2.39·37-s + 0.231·38-s + 0.160·40-s − 0.239·41-s + ⋯ |
| L(s) = 1 | − 0.0377·2-s − 0.998·4-s + 0.336·5-s + 0.942·7-s + 0.0754·8-s − 0.0126·10-s + 1.51·11-s − 0.442·13-s − 0.0355·14-s + 0.995·16-s − 1.75·17-s − 0.995·19-s − 0.335·20-s − 0.0573·22-s − 0.780·23-s − 0.887·25-s + 0.0166·26-s − 0.941·28-s + 0.429·29-s − 0.404·31-s − 0.112·32-s + 0.0663·34-s + 0.316·35-s − 0.393·37-s + 0.0375·38-s + 0.0253·40-s − 0.0374·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0533T + 2T^{2} \) |
| 5 | \( 1 - 0.751T + 5T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 - 5.03T + 11T^{2} \) |
| 13 | \( 1 + 1.59T + 13T^{2} \) |
| 17 | \( 1 + 7.25T + 17T^{2} \) |
| 19 | \( 1 + 4.33T + 19T^{2} \) |
| 23 | \( 1 + 3.74T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 + 2.25T + 31T^{2} \) |
| 37 | \( 1 + 2.39T + 37T^{2} \) |
| 41 | \( 1 + 0.239T + 41T^{2} \) |
| 43 | \( 1 - 5.91T + 43T^{2} \) |
| 47 | \( 1 + 0.831T + 47T^{2} \) |
| 53 | \( 1 + 3.11T + 53T^{2} \) |
| 59 | \( 1 + 2.31T + 59T^{2} \) |
| 61 | \( 1 + 5.74T + 61T^{2} \) |
| 67 | \( 1 + 7.75T + 67T^{2} \) |
| 71 | \( 1 + 7.51T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 0.660T + 79T^{2} \) |
| 83 | \( 1 + 1.59T + 83T^{2} \) |
| 89 | \( 1 + 2.41T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
| 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
−8.268440597224707104978361122618, −7.45059664600156694431422636900, −6.47575889995172740084984891230, −5.92376496324615701834881186382, −4.80072430151440202999765732972, −4.38449574760792964872692634116, −3.72584303317225222524626678266, −2.23094796648728509567516524185, −1.47489642694450784402957965872, 0,
1.47489642694450784402957965872, 2.23094796648728509567516524185, 3.72584303317225222524626678266, 4.38449574760792964872692634116, 4.80072430151440202999765732972, 5.92376496324615701834881186382, 6.47575889995172740084984891230, 7.45059664600156694431422636900, 8.268440597224707104978361122618
