L(s) = 1 | − 1.46·2-s + 0.151·4-s − 0.466·5-s + 7-s + 2.71·8-s + 0.684·10-s + 1.58·13-s − 1.46·14-s − 4.27·16-s + 5.22·17-s − 4.22·19-s − 0.0706·20-s + 1.80·23-s − 4.78·25-s − 2.32·26-s + 0.151·28-s + 2.71·29-s + 1.29·31-s + 0.854·32-s − 7.66·34-s − 0.466·35-s − 1.94·37-s + 6.19·38-s − 1.26·40-s − 1.04·41-s − 8.70·43-s − 2.64·46-s + ⋯ |
L(s) = 1 | − 1.03·2-s + 0.0756·4-s − 0.208·5-s + 0.377·7-s + 0.958·8-s + 0.216·10-s + 0.438·13-s − 0.392·14-s − 1.06·16-s + 1.26·17-s − 0.968·19-s − 0.0157·20-s + 0.376·23-s − 0.956·25-s − 0.455·26-s + 0.0285·28-s + 0.504·29-s + 0.232·31-s + 0.150·32-s − 1.31·34-s − 0.0788·35-s − 0.319·37-s + 1.00·38-s − 0.200·40-s − 0.162·41-s − 1.32·43-s − 0.390·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.015611749\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.015611749\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + 1.46T + 2T^{2} \) |
| 5 | \( 1 + 0.466T + 5T^{2} \) |
| 13 | \( 1 - 1.58T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 + 4.22T + 19T^{2} \) |
| 23 | \( 1 - 1.80T + 23T^{2} \) |
| 29 | \( 1 - 2.71T + 29T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 37 | \( 1 + 1.94T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 + 8.70T + 43T^{2} \) |
| 47 | \( 1 - 6.39T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 - 8.60T + 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 + 4.67T + 67T^{2} \) |
| 71 | \( 1 + 9.74T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + 3.58T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 - 8.91T + 89T^{2} \) |
| 97 | \( 1 + 2.70T + 97T^{2} \) |
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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
−8.170906607247900245027817422442, −7.33438694142328478279250944797, −6.79109306299082756095942333973, −5.76591842040212741778025634993, −5.12806556694172756667426358838, −4.21576505546798691881247878998, −3.63924377940535851629912829210, −2.42843299167210931775736701033, −1.50212543390285906280860303624, −0.63627559022666469651044724620,
0.63627559022666469651044724620, 1.50212543390285906280860303624, 2.42843299167210931775736701033, 3.63924377940535851629912829210, 4.21576505546798691881247878998, 5.12806556694172756667426358838, 5.76591842040212741778025634993, 6.79109306299082756095942333973, 7.33438694142328478279250944797, 8.170906607247900245027817422442