L(s) = 1 | − 2.41·2-s + 3.82·4-s − 5-s + 1.41·7-s − 4.41·8-s + 2.41·10-s − 4·11-s − 0.828·13-s − 3.41·14-s + 2.99·16-s − 6.82·17-s − 2.24·19-s − 3.82·20-s + 9.65·22-s + 8.24·23-s + 25-s + 1.99·26-s + 5.41·28-s + 8.82·29-s + 3.41·31-s + 1.58·32-s + 16.4·34-s − 1.41·35-s − 3.65·37-s + 5.41·38-s + 4.41·40-s + 4.82·41-s + ⋯ |
L(s) = 1 | − 1.70·2-s + 1.91·4-s − 0.447·5-s + 0.534·7-s − 1.56·8-s + 0.763·10-s − 1.20·11-s − 0.229·13-s − 0.912·14-s + 0.749·16-s − 1.65·17-s − 0.514·19-s − 0.856·20-s + 2.05·22-s + 1.71·23-s + 0.200·25-s + 0.392·26-s + 1.02·28-s + 1.63·29-s + 0.613·31-s + 0.280·32-s + 2.82·34-s − 0.239·35-s − 0.601·37-s + 0.878·38-s + 0.697·40-s + 0.754·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 89 | \( 1 - T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 + 6.82T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 - 8.24T + 23T^{2} \) |
| 29 | \( 1 - 8.82T + 29T^{2} \) |
| 31 | \( 1 - 3.41T + 31T^{2} \) |
| 37 | \( 1 + 3.65T + 37T^{2} \) |
| 41 | \( 1 - 4.82T + 41T^{2} \) |
| 43 | \( 1 - 7.07T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 + 0.928T + 83T^{2} \) |
| 97 | \( 1 - 2.82T + 97T^{2} \) |
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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
−8.185710814784265869329006321284, −7.60594993277371331378059907316, −6.93100095961537812758274569961, −6.28386557297579558006906324522, −4.98661449710015688508073236816, −4.43680478975301279775087709259, −2.85124709653380150914359845643, −2.32609617004920733328924788507, −1.08671725308521691673946371404, 0,
1.08671725308521691673946371404, 2.32609617004920733328924788507, 2.85124709653380150914359845643, 4.43680478975301279775087709259, 4.98661449710015688508073236816, 6.28386557297579558006906324522, 6.93100095961537812758274569961, 7.60594993277371331378059907316, 8.185710814784265869329006321284