| L(s) = 1 | + 0.467·2-s − 1.78·4-s + 0.0105·5-s − 1.85·7-s − 1.76·8-s + 0.00492·10-s + 1.01·11-s + 2.69·13-s − 0.868·14-s + 2.73·16-s + 3.50·17-s − 7.48·19-s − 0.0187·20-s + 0.472·22-s − 23-s − 4.99·25-s + 1.26·26-s + 3.31·28-s + 29-s + 9.06·31-s + 4.81·32-s + 1.63·34-s − 0.0195·35-s − 3.80·37-s − 3.49·38-s − 0.0186·40-s − 0.921·41-s + ⋯ |
| L(s) = 1 | + 0.330·2-s − 0.890·4-s + 0.00471·5-s − 0.702·7-s − 0.624·8-s + 0.00155·10-s + 0.305·11-s + 0.748·13-s − 0.232·14-s + 0.684·16-s + 0.851·17-s − 1.71·19-s − 0.00420·20-s + 0.100·22-s − 0.208·23-s − 0.999·25-s + 0.247·26-s + 0.626·28-s + 0.185·29-s + 1.62·31-s + 0.850·32-s + 0.281·34-s − 0.00331·35-s − 0.625·37-s − 0.566·38-s − 0.00294·40-s − 0.143·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 0.467T + 2T^{2} \) |
| 5 | \( 1 - 0.0105T + 5T^{2} \) |
| 7 | \( 1 + 1.85T + 7T^{2} \) |
| 11 | \( 1 - 1.01T + 11T^{2} \) |
| 13 | \( 1 - 2.69T + 13T^{2} \) |
| 17 | \( 1 - 3.50T + 17T^{2} \) |
| 19 | \( 1 + 7.48T + 19T^{2} \) |
| 31 | \( 1 - 9.06T + 31T^{2} \) |
| 37 | \( 1 + 3.80T + 37T^{2} \) |
| 41 | \( 1 + 0.921T + 41T^{2} \) |
| 43 | \( 1 - 5.98T + 43T^{2} \) |
| 47 | \( 1 - 8.75T + 47T^{2} \) |
| 53 | \( 1 - 9.78T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 + 1.50T + 61T^{2} \) |
| 67 | \( 1 + 7.98T + 67T^{2} \) |
| 71 | \( 1 + 3.19T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 9.57T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 + 9.07T + 89T^{2} \) |
| 97 | \( 1 - 1.28T + 97T^{2} \) |
| 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
−7.916364920955223046725467937541, −6.85648915517481861237893870138, −6.06495889541878464925401047492, −5.78600292863150342650482017475, −4.64611689002030465574977635936, −4.06070050795120900391071907274, −3.44726705868363189815488962538, −2.50488563267297068479498272611, −1.18090658188474038834695081915, 0,
1.18090658188474038834695081915, 2.50488563267297068479498272611, 3.44726705868363189815488962538, 4.06070050795120900391071907274, 4.64611689002030465574977635936, 5.78600292863150342650482017475, 6.06495889541878464925401047492, 6.85648915517481861237893870138, 7.916364920955223046725467937541
