| L(s) = 1 | − 4·2-s − 17.4·3-s + 16·4-s + 69.9·6-s + 49·7-s − 64·8-s + 62.4·9-s − 282.·11-s − 279.·12-s + 823.·13-s − 196·14-s + 256·16-s + 2.18e3·17-s − 249.·18-s − 2.83e3·19-s − 856.·21-s + 1.13e3·22-s − 2.70e3·23-s + 1.11e3·24-s − 3.29e3·26-s + 3.15e3·27-s + 784·28-s + 3.64e3·29-s + 378.·31-s − 1.02e3·32-s + 4.93e3·33-s − 8.72e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.12·3-s + 0.5·4-s + 0.792·6-s + 0.377·7-s − 0.353·8-s + 0.256·9-s − 0.704·11-s − 0.560·12-s + 1.35·13-s − 0.267·14-s + 0.250·16-s + 1.83·17-s − 0.181·18-s − 1.80·19-s − 0.423·21-s + 0.497·22-s − 1.06·23-s + 0.396·24-s − 0.955·26-s + 0.833·27-s + 0.188·28-s + 0.803·29-s + 0.0706·31-s − 0.176·32-s + 0.789·33-s − 1.29·34-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
| \(L(3)\) |
\(\approx\) |
\(0.7575906564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7575906564\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 17.4T + 243T^{2} \) |
| 11 | \( 1 + 282.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 823.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.18e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.83e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.70e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.64e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 378.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.38e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.90e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.34e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.67e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.76e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.17e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.24e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.99e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.94e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.51e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.68e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.01e4T + 8.58e9T^{2} \) |
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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
−10.43537110371411630445461394527, −10.23379438035019867045325086458, −8.525546228106927597543392936698, −8.111096067724709015515401464520, −6.67057707159578511254936764734, −5.93677754939708575309419079029, −5.00733325988676486485927909554, −3.46765311068902057194607879599, −1.78535808752802243696603999293, −0.56237064475606475663183424170,
0.56237064475606475663183424170, 1.78535808752802243696603999293, 3.46765311068902057194607879599, 5.00733325988676486485927909554, 5.93677754939708575309419079029, 6.67057707159578511254936764734, 8.111096067724709015515401464520, 8.525546228106927597543392936698, 10.23379438035019867045325086458, 10.43537110371411630445461394527
