| L(s) = 1 | − 1.86·2-s − 9.53·3-s − 4.53·4-s − 5·5-s + 17.7·6-s + 23.3·8-s + 63.9·9-s + 9.30·10-s − 36.9·11-s + 43.2·12-s + 22.7·13-s + 47.6·15-s − 7.13·16-s + 135.·17-s − 119.·18-s − 6.22·19-s + 22.6·20-s + 68.8·22-s − 48.7·23-s − 222.·24-s + 25·25-s − 42.4·26-s − 352.·27-s − 71.1·29-s − 88.7·30-s − 124.·31-s − 173.·32-s + ⋯ |
| L(s) = 1 | − 0.657·2-s − 1.83·3-s − 0.567·4-s − 0.447·5-s + 1.20·6-s + 1.03·8-s + 2.36·9-s + 0.294·10-s − 1.01·11-s + 1.04·12-s + 0.486·13-s + 0.820·15-s − 0.111·16-s + 1.93·17-s − 1.55·18-s − 0.0751·19-s + 0.253·20-s + 0.666·22-s − 0.441·23-s − 1.89·24-s + 0.200·25-s − 0.319·26-s − 2.51·27-s − 0.455·29-s − 0.540·30-s − 0.723·31-s − 0.957·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.86T + 8T^{2} \) |
| 3 | \( 1 + 9.53T + 27T^{2} \) |
| 11 | \( 1 + 36.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 22.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 135.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.22T + 6.85e3T^{2} \) |
| 23 | \( 1 + 48.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 71.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 84.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 92.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 299.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 72.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 362.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 375.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 689.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 972.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 281.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 742.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 592.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 493.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 962.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 740.T + 9.12e5T^{2} \) |
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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
−10.92493358846457950603867329325, −10.41395946259659566835164556043, −9.525361657904853587792311054206, −8.019728696666361815861907341690, −7.31276585074945345600819315973, −5.81665524176866826002497594441, −5.14343273643395820921210671017, −3.93618494404550254776334371517, −1.12835310643349673566618977629, 0,
1.12835310643349673566618977629, 3.93618494404550254776334371517, 5.14343273643395820921210671017, 5.81665524176866826002497594441, 7.31276585074945345600819315973, 8.019728696666361815861907341690, 9.525361657904853587792311054206, 10.41395946259659566835164556043, 10.92493358846457950603867329325
