| L(s) = 1 | + 4·2-s − 27.7·3-s + 16·4-s − 111.·6-s + 49·7-s + 64·8-s + 528.·9-s − 481.·11-s − 444.·12-s − 883.·13-s + 196·14-s + 256·16-s + 1.84e3·17-s + 2.11e3·18-s + 2.44e3·19-s − 1.36e3·21-s − 1.92e3·22-s − 169.·23-s − 1.77e3·24-s − 3.53e3·26-s − 7.93e3·27-s + 784·28-s − 554.·29-s + 7.73e3·31-s + 1.02e3·32-s + 1.33e4·33-s + 7.36e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.78·3-s + 0.5·4-s − 1.25·6-s + 0.377·7-s + 0.353·8-s + 2.17·9-s − 1.20·11-s − 0.890·12-s − 1.44·13-s + 0.267·14-s + 0.250·16-s + 1.54·17-s + 1.53·18-s + 1.55·19-s − 0.673·21-s − 0.848·22-s − 0.0667·23-s − 0.629·24-s − 1.02·26-s − 2.09·27-s + 0.188·28-s − 0.122·29-s + 1.44·31-s + 0.176·32-s + 2.13·33-s + 1.09·34-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 | \( 1 + 27.7T + 243T^{2} \) |
| 11 | \( 1 + 481.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 883.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.84e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.44e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 169.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 554.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.08e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.53e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.70e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 8.82e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.05e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.67e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 9.88e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.67e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.86e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.17e5T + 8.58e9T^{2} \) |
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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
−10.27540973305645128906272649305, −9.932932255961855880138177266112, −7.76536059268431391030723163386, −7.20671086700556534058337173807, −5.92406803849994200230618472770, −5.18507431828783265117325603195, −4.74071531561941309969167620057, −3.01517159164816602782255588505, −1.32525086143164951294428904605, 0,
1.32525086143164951294428904605, 3.01517159164816602782255588505, 4.74071531561941309969167620057, 5.18507431828783265117325603195, 5.92406803849994200230618472770, 7.20671086700556534058337173807, 7.76536059268431391030723163386, 9.932932255961855880138177266112, 10.27540973305645128906272649305
