| L(s) = 1 | + 4·2-s − 19.0·3-s + 16·4-s − 76.0·6-s − 49·7-s + 64·8-s + 118.·9-s + 539.·11-s − 304.·12-s − 554.·13-s − 196·14-s + 256·16-s + 934.·17-s + 472.·18-s − 600.·19-s + 931.·21-s + 2.15e3·22-s − 1.76e3·23-s − 1.21e3·24-s − 2.21e3·26-s + 2.37e3·27-s − 784·28-s + 5.70e3·29-s + 1.99e3·31-s + 1.02e3·32-s − 1.02e4·33-s + 3.73e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.21·3-s + 0.5·4-s − 0.862·6-s − 0.377·7-s + 0.353·8-s + 0.486·9-s + 1.34·11-s − 0.609·12-s − 0.910·13-s − 0.267·14-s + 0.250·16-s + 0.784·17-s + 0.343·18-s − 0.381·19-s + 0.460·21-s + 0.950·22-s − 0.697·23-s − 0.431·24-s − 0.643·26-s + 0.626·27-s − 0.188·28-s + 1.25·29-s + 0.373·31-s + 0.176·32-s − 1.63·33-s + 0.554·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 4T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 + 19.0T + 243T^{2} \) |
| 11 | \( 1 - 539.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 554.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 934.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 600.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.76e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.70e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.99e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 550.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.18e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.25e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.58e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.04e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.23e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.38e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.50e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.41e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.40e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.18e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.58e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.77e4T + 8.58e9T^{2} \) |
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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
−10.37158209572590645843542506739, −9.589571349755259059974176430838, −8.154100805234165342657733351108, −6.75148513210617761404703589701, −6.33491900211457351530016372233, −5.25176414098540650230318464210, −4.37477546671816255131885037314, −3.10127179613868825710306228844, −1.43128421211971704360131215529, 0,
1.43128421211971704360131215529, 3.10127179613868825710306228844, 4.37477546671816255131885037314, 5.25176414098540650230318464210, 6.33491900211457351530016372233, 6.75148513210617761404703589701, 8.154100805234165342657733351108, 9.589571349755259059974176430838, 10.37158209572590645843542506739
