| L(s) = 1 | − 2.21·2-s + 5.10·3-s − 27.0·4-s − 25·5-s − 11.3·6-s + 167.·7-s + 130.·8-s − 216.·9-s + 55.4·10-s − 121·11-s − 138.·12-s − 694.·13-s − 370.·14-s − 127.·15-s + 576.·16-s − 1.05e3·17-s + 480.·18-s + 361·19-s + 677.·20-s + 853.·21-s + 268.·22-s + 297.·23-s + 669.·24-s + 625·25-s + 1.53e3·26-s − 2.34e3·27-s − 4.52e3·28-s + ⋯ |
| L(s) = 1 | − 0.391·2-s + 0.327·3-s − 0.846·4-s − 0.447·5-s − 0.128·6-s + 1.28·7-s + 0.723·8-s − 0.892·9-s + 0.175·10-s − 0.301·11-s − 0.277·12-s − 1.13·13-s − 0.504·14-s − 0.146·15-s + 0.562·16-s − 0.885·17-s + 0.349·18-s + 0.229·19-s + 0.378·20-s + 0.422·21-s + 0.118·22-s + 0.117·23-s + 0.237·24-s + 0.200·25-s + 0.446·26-s − 0.620·27-s − 1.09·28-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
| \(L(3)\) |
\(\approx\) |
\(0.8029344676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8029344676\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 2.21T + 32T^{2} \) |
| 3 | \( 1 - 5.10T + 243T^{2} \) |
| 7 | \( 1 - 167.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 694.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.05e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 297.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.32e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.93e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.08e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.76e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.57e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 4.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.20e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.53e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.30e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.47e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.10e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.64e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.90e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.51e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.21e4T + 8.58e9T^{2} \) |
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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
−8.998869808843445206332797890995, −8.337127967461461313931835274827, −7.88146986865786805197438790954, −6.98953987462834193039855154047, −5.43894759260845072516157380231, −4.87963738912424223116629021883, −4.05268165582556225096352788059, −2.79628780343688180472128615697, −1.72080660821700309066108617162, −0.40408748726911701344603014294,
0.40408748726911701344603014294, 1.72080660821700309066108617162, 2.79628780343688180472128615697, 4.05268165582556225096352788059, 4.87963738912424223116629021883, 5.43894759260845072516157380231, 6.98953987462834193039855154047, 7.88146986865786805197438790954, 8.337127967461461313931835274827, 8.998869808843445206332797890995
