| L(s) = 1 | − 2·2-s + 7.78·3-s + 4·4-s − 5·5-s − 15.5·6-s − 8·8-s + 33.5·9-s + 10·10-s − 47.5·11-s + 31.1·12-s − 1.12·13-s − 38.9·15-s + 16·16-s − 136.·17-s − 67.1·18-s + 88.8·19-s − 20·20-s + 95.1·22-s + 2.73·23-s − 62.2·24-s + 25·25-s + 2.25·26-s + 51.0·27-s − 131.·29-s + 77.8·30-s − 301.·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.49·3-s + 0.5·4-s − 0.447·5-s − 1.05·6-s − 0.353·8-s + 1.24·9-s + 0.316·10-s − 1.30·11-s + 0.748·12-s − 0.0240·13-s − 0.669·15-s + 0.250·16-s − 1.94·17-s − 0.879·18-s + 1.07·19-s − 0.223·20-s + 0.921·22-s + 0.0247·23-s − 0.529·24-s + 0.200·25-s + 0.0170·26-s + 0.364·27-s − 0.839·29-s + 0.473·30-s − 1.74·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 7.78T + 27T^{2} \) |
| 11 | \( 1 + 47.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 1.12T + 2.19e3T^{2} \) |
| 17 | \( 1 + 136.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 2.73T + 1.21e4T^{2} \) |
| 29 | \( 1 + 131.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 301.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 288.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 70.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 136.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 34.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 745.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 683.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 72.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 687.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 84.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 722.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.01e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 424.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 120.T + 9.12e5T^{2} \) |
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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
−9.834004188612689274553040635034, −9.045259724049241437119985722194, −8.386456487273605606709753321467, −7.61973981069178400487584782718, −6.94995312692386906741403684678, −5.30683031437977801905639643482, −3.89902629033933013211748808005, −2.83608812274789522453266314827, −1.94779501270488225540316155032, 0,
1.94779501270488225540316155032, 2.83608812274789522453266314827, 3.89902629033933013211748808005, 5.30683031437977801905639643482, 6.94995312692386906741403684678, 7.61973981069178400487584782718, 8.386456487273605606709753321467, 9.045259724049241437119985722194, 9.834004188612689274553040635034
