| L(s) = 1 | − 7.16·2-s − 23.9·3-s + 19.3·4-s − 25·5-s + 171.·6-s − 233.·7-s + 90.8·8-s + 331.·9-s + 179.·10-s + 121·11-s − 462.·12-s − 389.·13-s + 1.67e3·14-s + 599.·15-s − 1.26e3·16-s − 1.66e3·17-s − 2.37e3·18-s − 361·19-s − 482.·20-s + 5.60e3·21-s − 866.·22-s + 3.69e3·23-s − 2.17e3·24-s + 625·25-s + 2.79e3·26-s − 2.11e3·27-s − 4.51e3·28-s + ⋯ |
| L(s) = 1 | − 1.26·2-s − 1.53·3-s + 0.603·4-s − 0.447·5-s + 1.94·6-s − 1.80·7-s + 0.501·8-s + 1.36·9-s + 0.566·10-s + 0.301·11-s − 0.928·12-s − 0.639·13-s + 2.28·14-s + 0.687·15-s − 1.23·16-s − 1.39·17-s − 1.72·18-s − 0.229·19-s − 0.269·20-s + 2.77·21-s − 0.381·22-s + 1.45·23-s − 0.771·24-s + 0.200·25-s + 0.809·26-s − 0.559·27-s − 1.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.01604430644\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01604430644\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 + 7.16T + 32T^{2} \) |
| 3 | \( 1 + 23.9T + 243T^{2} \) |
| 7 | \( 1 + 233.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 389.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.66e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.42e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.22e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.93e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.16e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.21e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.86e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.91e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.48e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.56e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.44e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.82e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.43e4T + 8.58e9T^{2} \) |
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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
−9.429883727091622080214606914744, −8.583641110186388954717892249815, −7.24365109791881619835591024732, −6.82389514848631946920491745801, −6.13952912071441966879704223388, −4.95138103379244324569732014624, −4.06456118464857119344558847050, −2.65517307365831202964344564646, −1.09444314581816974969828417196, −0.087694539053941556883872902035,
0.087694539053941556883872902035, 1.09444314581816974969828417196, 2.65517307365831202964344564646, 4.06456118464857119344558847050, 4.95138103379244324569732014624, 6.13952912071441966879704223388, 6.82389514848631946920491745801, 7.24365109791881619835591024732, 8.583641110186388954717892249815, 9.429883727091622080214606914744
