| L(s) = 1 | − 4.70·2-s + 14.1·4-s + 5·5-s + 7·7-s − 28.7·8-s − 23.5·10-s − 24.5·11-s − 35.0·13-s − 32.9·14-s + 22.1·16-s + 18.4·17-s − 67.4·19-s + 70.5·20-s + 115.·22-s + 145.·23-s + 25·25-s + 164.·26-s + 98.7·28-s − 214.·29-s − 88.6·31-s + 125.·32-s − 86.5·34-s + 35·35-s + 162.·37-s + 316.·38-s − 143.·40-s + 337.·41-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 1.76·4-s + 0.447·5-s + 0.377·7-s − 1.26·8-s − 0.743·10-s − 0.674·11-s − 0.747·13-s − 0.628·14-s + 0.345·16-s + 0.262·17-s − 0.813·19-s + 0.788·20-s + 1.12·22-s + 1.32·23-s + 0.200·25-s + 1.24·26-s + 0.666·28-s − 1.37·29-s − 0.513·31-s + 0.694·32-s − 0.436·34-s + 0.169·35-s + 0.720·37-s + 1.35·38-s − 0.567·40-s + 1.28·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 4.70T + 8T^{2} \) |
| 11 | \( 1 + 24.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 35.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 18.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 67.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 88.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 162.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 122.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 676.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 501.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 708.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 907.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 430.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 41.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 890.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.47e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 555.T + 9.12e5T^{2} \) |
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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
−10.73367796985506401003902973602, −9.589166291319292280483701314920, −9.088519870035481934270409011195, −7.919320767137666454790690186965, −7.34645259867073888196785873983, −6.11845432029063796772736407004, −4.79606338446240584283964420960, −2.72285383104770266038427571937, −1.56122235597130864247503891068, 0,
1.56122235597130864247503891068, 2.72285383104770266038427571937, 4.79606338446240584283964420960, 6.11845432029063796772736407004, 7.34645259867073888196785873983, 7.919320767137666454790690186965, 9.088519870035481934270409011195, 9.589166291319292280483701314920, 10.73367796985506401003902973602
