| L(s) = 1 | − 307. i·5-s + (222. − 2.39e3i)7-s − 5.38e3·11-s + 4.89e4i·13-s − 5.54e4i·17-s − 6.77e4i·19-s + 3.44e5·23-s + 2.96e5·25-s + 1.17e6·29-s − 1.25e5i·31-s + (−7.34e5 − 6.82e4i)35-s − 7.34e5·37-s − 1.18e6i·41-s − 3.84e6·43-s + 4.82e6i·47-s + ⋯ |
| L(s) = 1 | − 0.491i·5-s + (0.0925 − 0.995i)7-s − 0.367·11-s + 1.71i·13-s − 0.663i·17-s − 0.520i·19-s + 1.23·23-s + 0.758·25-s + 1.65·29-s − 0.136i·31-s + (−0.489 − 0.0454i)35-s − 0.391·37-s − 0.420i·41-s − 1.12·43-s + 0.989i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0925 + 0.995i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0925 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.885597853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.885597853\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-222. + 2.39e3i)T \) |
| good | 5 | \( 1 + 307. iT - 3.90e5T^{2} \) |
| 11 | \( 1 + 5.38e3T + 2.14e8T^{2} \) |
| 13 | \( 1 - 4.89e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 5.54e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 6.77e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 3.44e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 1.17e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.25e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 7.34e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 1.18e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 3.84e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 4.82e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.19e7T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.90e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.11e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 2.92e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 2.79e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + 3.34e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 4.96e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + 3.33e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 3.09e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 5.05e7iT - 7.83e15T^{2} \) |
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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
−10.39971759418374746588948891902, −9.319664467285097780012490593658, −8.549575357099553092540543825421, −7.23356099895899382356723207570, −6.63433808060440563059727180585, −4.96296546196556810567733416128, −4.37708634796960004254518927060, −2.95155328580253839359918339424, −1.49216885378794838821131538181, −0.47116182001538848507232076734,
1.03324897832113609116950078097, 2.56395256792685622195629346949, 3.27917633005718784259871157829, 4.97424215460780855540177130170, 5.78032174700191912317306948158, 6.87954979117002673440615068823, 8.117533876790863250056983300605, 8.742075267478214074517777945965, 10.17686970195406244082016870993, 10.65394645156411065980835491494
