| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s + 2.42·7-s − 8-s + 9-s − 10-s − 11-s − 12-s + 1.11·13-s − 2.42·14-s − 15-s + 16-s + 17-s − 18-s + 0.421·19-s + 20-s − 2.42·21-s + 22-s − 5.96·23-s + 24-s + 25-s − 1.11·26-s − 27-s + 2.42·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 0.915·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s + 0.310·13-s − 0.647·14-s − 0.258·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.0967·19-s + 0.223·20-s − 0.528·21-s + 0.213·22-s − 1.24·23-s + 0.204·24-s + 0.200·25-s − 0.219·26-s − 0.192·27-s + 0.457·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2.42T + 7T^{2} \) |
| 13 | \( 1 - 1.11T + 13T^{2} \) |
| 19 | \( 1 - 0.421T + 19T^{2} \) |
| 23 | \( 1 + 5.96T + 23T^{2} \) |
| 29 | \( 1 - 6.57T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 6.51T + 37T^{2} \) |
| 41 | \( 1 + 4.84T + 41T^{2} \) |
| 43 | \( 1 - 7.59T + 43T^{2} \) |
| 47 | \( 1 + 3.73T + 47T^{2} \) |
| 53 | \( 1 + 5.82T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 0.134T + 61T^{2} \) |
| 67 | \( 1 + 0.978T + 67T^{2} \) |
| 71 | \( 1 + 6.74T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 9.59T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 1.08T + 89T^{2} \) |
| 97 | \( 1 + 7.55T + 97T^{2} \) |
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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
−7.84604741906000471554268320037, −7.17047415118657323506042616423, −6.36416709538143439543833217765, −5.67960620921647646851454692187, −5.06839171986064613232782194933, −4.18097866820459678706433041273, −3.11628490451257282004335490414, −1.95667895141289558191209022630, −1.37838481067971724126154070517, 0,
1.37838481067971724126154070517, 1.95667895141289558191209022630, 3.11628490451257282004335490414, 4.18097866820459678706433041273, 5.06839171986064613232782194933, 5.67960620921647646851454692187, 6.36416709538143439543833217765, 7.17047415118657323506042616423, 7.84604741906000471554268320037
