| L(s) = 1 | + 25·5-s − 33.1·7-s − 137.·11-s − 227.·13-s − 830.·17-s + 270.·19-s + 1.10e3·23-s + 625·25-s + 8.77e3·29-s + 5.41e3·31-s − 827.·35-s + 4.94e3·37-s − 1.39e4·41-s − 1.16e4·43-s − 1.09e4·47-s − 1.57e4·49-s − 9.17e3·53-s − 3.44e3·55-s − 1.90e4·59-s + 4.24e4·61-s − 5.69e3·65-s + 2.86e4·67-s + 4.99e4·71-s − 5.49e4·73-s + 4.56e3·77-s − 2.35e4·79-s − 3.07e4·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.255·7-s − 0.343·11-s − 0.374·13-s − 0.696·17-s + 0.171·19-s + 0.434·23-s + 0.200·25-s + 1.93·29-s + 1.01·31-s − 0.114·35-s + 0.594·37-s − 1.29·41-s − 0.964·43-s − 0.724·47-s − 0.934·49-s − 0.448·53-s − 0.153·55-s − 0.711·59-s + 1.46·61-s − 0.167·65-s + 0.779·67-s + 1.17·71-s − 1.20·73-s + 0.0877·77-s − 0.424·79-s − 0.489·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 7 | \( 1 + 33.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 137.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 227.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 830.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 270.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.41e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.94e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.39e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.16e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.09e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.17e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.90e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.24e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.99e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.49e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.35e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.07e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.68e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.38e5T + 8.58e9T^{2} \) |
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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
−8.665283990390308726057694466862, −8.062642313002520131683410809044, −6.83596545044430436454000020832, −6.39315399513127543367608379827, −5.18956978029216200526922975670, −4.55425117733215234143888042924, −3.20838920354322821996047174187, −2.41415708012923262405815595388, −1.21216488252169566402675192787, 0,
1.21216488252169566402675192787, 2.41415708012923262405815595388, 3.20838920354322821996047174187, 4.55425117733215234143888042924, 5.18956978029216200526922975670, 6.39315399513127543367608379827, 6.83596545044430436454000020832, 8.062642313002520131683410809044, 8.665283990390308726057694466862
