L(s) = 1 | + 2-s − 0.489·3-s + 4-s + 5-s − 0.489·6-s − 2.03·7-s + 8-s − 2.76·9-s + 10-s + 3.55·11-s − 0.489·12-s − 6.34·13-s − 2.03·14-s − 0.489·15-s + 16-s − 0.509·17-s − 2.76·18-s + 6.21·19-s + 20-s + 0.995·21-s + 3.55·22-s + 0.0973·23-s − 0.489·24-s + 25-s − 6.34·26-s + 2.81·27-s − 2.03·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.282·3-s + 0.5·4-s + 0.447·5-s − 0.199·6-s − 0.769·7-s + 0.353·8-s − 0.920·9-s + 0.316·10-s + 1.07·11-s − 0.141·12-s − 1.76·13-s − 0.544·14-s − 0.126·15-s + 0.250·16-s − 0.123·17-s − 0.650·18-s + 1.42·19-s + 0.223·20-s + 0.217·21-s + 0.758·22-s + 0.0203·23-s − 0.0998·24-s + 0.200·25-s − 1.24·26-s + 0.542·27-s − 0.384·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.509002108\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.509002108\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 - T \) |
good | 3 | \( 1 + 0.489T + 3T^{2} \) |
| 7 | \( 1 + 2.03T + 7T^{2} \) |
| 11 | \( 1 - 3.55T + 11T^{2} \) |
| 13 | \( 1 + 6.34T + 13T^{2} \) |
| 17 | \( 1 + 0.509T + 17T^{2} \) |
| 19 | \( 1 - 6.21T + 19T^{2} \) |
| 23 | \( 1 - 0.0973T + 23T^{2} \) |
| 29 | \( 1 - 0.202T + 29T^{2} \) |
| 31 | \( 1 - 8.32T + 31T^{2} \) |
| 37 | \( 1 + 4.09T + 37T^{2} \) |
| 41 | \( 1 + 5.06T + 41T^{2} \) |
| 43 | \( 1 + 6.14T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 4.00T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 - 11.8T + 61T^{2} \) |
| 67 | \( 1 - 0.114T + 67T^{2} \) |
| 71 | \( 1 - 5.12T + 71T^{2} \) |
| 73 | \( 1 + 3.80T + 73T^{2} \) |
| 79 | \( 1 - 9.23T + 79T^{2} \) |
| 83 | \( 1 - 2.73T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 - 5.74T + 97T^{2} \) |
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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
−7.958634060530132189069276625675, −6.98689934395737667357350348327, −6.68700794366390435809010325917, −5.84960292489903179824928277888, −5.23418662168586297276855495002, −4.62739744946484755358258433385, −3.52169312993020262894704386704, −2.91540247706945844957504218219, −2.10719679364039466274171742309, −0.73927043648813485228955701521,
0.73927043648813485228955701521, 2.10719679364039466274171742309, 2.91540247706945844957504218219, 3.52169312993020262894704386704, 4.62739744946484755358258433385, 5.23418662168586297276855495002, 5.84960292489903179824928277888, 6.68700794366390435809010325917, 6.98689934395737667357350348327, 7.958634060530132189069276625675