| L(s) = 1 | − 0.540·2-s − 7.70·4-s − 5·5-s + 24.4·7-s + 8.49·8-s + 2.70·10-s + 11·11-s − 84.5·13-s − 13.2·14-s + 57.0·16-s + 62.8·17-s − 159.·19-s + 38.5·20-s − 5.94·22-s + 114.·23-s + 25·25-s + 45.7·26-s − 188.·28-s − 172.·29-s + 8.87·31-s − 98.7·32-s − 33.9·34-s − 122.·35-s − 14.9·37-s + 86.0·38-s − 42.4·40-s + 463.·41-s + ⋯ |
| L(s) = 1 | − 0.191·2-s − 0.963·4-s − 0.447·5-s + 1.32·7-s + 0.375·8-s + 0.0854·10-s + 0.301·11-s − 1.80·13-s − 0.252·14-s + 0.891·16-s + 0.895·17-s − 1.92·19-s + 0.430·20-s − 0.0576·22-s + 1.03·23-s + 0.200·25-s + 0.344·26-s − 1.27·28-s − 1.10·29-s + 0.0513·31-s − 0.545·32-s − 0.171·34-s − 0.590·35-s − 0.0663·37-s + 0.367·38-s − 0.167·40-s + 1.76·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.239802986\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.239802986\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 0.540T + 8T^{2} \) |
| 7 | \( 1 - 24.4T + 343T^{2} \) |
| 13 | \( 1 + 84.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 62.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 159.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 172.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 8.87T + 2.97e4T^{2} \) |
| 37 | \( 1 + 14.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 463.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 486.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 118.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 273.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 884.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 347.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 720.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 71.7T + 3.57e5T^{2} \) |
| 73 | \( 1 - 146.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 147.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 399.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.64e3T + 9.12e5T^{2} \) |
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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
−10.54598261929083925152630593656, −9.524953763233663316335457722816, −8.730352624436293283153235845596, −7.86630856350660968106241047604, −7.24731516149416053850335906283, −5.56686120269113420931675570709, −4.71088217857371225535173219391, −4.01922771926169342417779860330, −2.27041788508577905101589228750, −0.72386003351777221329193346801,
0.72386003351777221329193346801, 2.27041788508577905101589228750, 4.01922771926169342417779860330, 4.71088217857371225535173219391, 5.56686120269113420931675570709, 7.24731516149416053850335906283, 7.86630856350660968106241047604, 8.730352624436293283153235845596, 9.524953763233663316335457722816, 10.54598261929083925152630593656
