| L(s) = 1 | + 3.12·2-s − 6.47·3-s − 22.2·4-s − 25·5-s − 20.2·6-s − 46.8·7-s − 169.·8-s − 201.·9-s − 78.0·10-s + 144.·12-s + 924.·13-s − 146.·14-s + 161.·15-s + 183.·16-s + 2.08e3·17-s − 627.·18-s − 663.·19-s + 556.·20-s + 303.·21-s − 753.·23-s + 1.09e3·24-s + 625·25-s + 2.88e3·26-s + 2.87e3·27-s + 1.04e3·28-s − 587.·29-s + 505.·30-s + ⋯ |
| L(s) = 1 | + 0.551·2-s − 0.415·3-s − 0.695·4-s − 0.447·5-s − 0.229·6-s − 0.361·7-s − 0.935·8-s − 0.827·9-s − 0.246·10-s + 0.289·12-s + 1.51·13-s − 0.199·14-s + 0.185·15-s + 0.179·16-s + 1.75·17-s − 0.456·18-s − 0.421·19-s + 0.311·20-s + 0.150·21-s − 0.296·23-s + 0.388·24-s + 0.200·25-s + 0.837·26-s + 0.759·27-s + 0.251·28-s − 0.129·29-s + 0.102·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 3.12T + 32T^{2} \) |
| 3 | \( 1 + 6.47T + 243T^{2} \) |
| 7 | \( 1 + 46.8T + 1.68e4T^{2} \) |
| 13 | \( 1 - 924.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.08e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 663.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 753.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 587.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.23e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.51e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.83e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.26e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.69e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.21e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.36e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.79e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.63e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.40e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.42e4T + 8.58e9T^{2} \) |
| 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
−9.386602367513736502233381430540, −8.520501218070065971782934750830, −7.83793972361170612609552467887, −6.27298441573193978493948615326, −5.82945715470011230447749815781, −4.81193729282575780667890855418, −3.69341965883145223017989011535, −3.08548991167124009181712004319, −1.07992856531623303727493413947, 0,
1.07992856531623303727493413947, 3.08548991167124009181712004319, 3.69341965883145223017989011535, 4.81193729282575780667890855418, 5.82945715470011230447749815781, 6.27298441573193978493948615326, 7.83793972361170612609552467887, 8.520501218070065971782934750830, 9.386602367513736502233381430540
