| L(s) = 1 | − 2·2-s + 3-s + 2·4-s − 5-s − 2·6-s + 0.561·7-s + 9-s + 2·10-s + 3.56·11-s + 2·12-s + 13-s − 1.12·14-s − 15-s − 4·16-s − 5.12·17-s − 2·18-s + 3.12·19-s − 2·20-s + 0.561·21-s − 7.12·22-s − 2·23-s + 25-s − 2·26-s + 27-s + 1.12·28-s − 7.68·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 0.212·7-s + 0.333·9-s + 0.632·10-s + 1.07·11-s + 0.577·12-s + 0.277·13-s − 0.300·14-s − 0.258·15-s − 16-s − 1.24·17-s − 0.471·18-s + 0.716·19-s − 0.447·20-s + 0.122·21-s − 1.51·22-s − 0.417·23-s + 0.200·25-s − 0.392·26-s + 0.192·27-s + 0.212·28-s − 1.42·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2T + 2T^{2} \) |
| 7 | \( 1 - 0.561T + 7T^{2} \) |
| 11 | \( 1 - 3.56T + 11T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 7.68T + 29T^{2} \) |
| 37 | \( 1 - 4.43T + 37T^{2} \) |
| 41 | \( 1 + 0.561T + 41T^{2} \) |
| 43 | \( 1 + 4.56T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 1.12T + 53T^{2} \) |
| 59 | \( 1 + 14.8T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 3.56T + 73T^{2} \) |
| 79 | \( 1 - 2.87T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 8.43T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−7.81390659559384531496956764071, −7.32652767550126600607224139791, −6.69254848788174993130524992947, −5.81960559415112823335975146349, −4.53928641480930806531085469726, −4.05860142802821898062857542392, −3.04610056666620931805520262379, −1.94735864886894356701717190994, −1.28344815388036838767343573906, 0,
1.28344815388036838767343573906, 1.94735864886894356701717190994, 3.04610056666620931805520262379, 4.05860142802821898062857542392, 4.53928641480930806531085469726, 5.81960559415112823335975146349, 6.69254848788174993130524992947, 7.32652767550126600607224139791, 7.81390659559384531496956764071
