| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 4·7-s − 8-s + 9-s − 10-s − 4·11-s − 12-s + 13-s + 4·14-s − 15-s + 16-s − 6·17-s − 18-s − 4·19-s + 20-s + 4·21-s + 4·22-s − 8·23-s + 24-s + 25-s − 26-s − 27-s − 4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.288·12-s + 0.277·13-s + 1.06·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.872·21-s + 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−17.22276078102789, −16.37127098002175, −15.98137521703121, −15.54627413017915, −15.19485946813402, −13.88332582616279, −13.61955071649153, −12.87036781954790, −12.54134497051284, −11.84119848451457, −11.06747741780096, −10.40847592831689, −10.18932717118914, −9.588141527789430, −8.848477951651315, −8.297934937154952, −7.538874634540524, −6.656695717765393, −6.330193433647911, −5.902946149105992, −4.898997520121558, −4.160764456432114, −3.123826640354512, −2.473322283112879, −1.638602503286876, 0, 0,
1.638602503286876, 2.473322283112879, 3.123826640354512, 4.160764456432114, 4.898997520121558, 5.902946149105992, 6.330193433647911, 6.656695717765393, 7.538874634540524, 8.297934937154952, 8.848477951651315, 9.588141527789430, 10.18932717118914, 10.40847592831689, 11.06747741780096, 11.84119848451457, 12.54134497051284, 12.87036781954790, 13.61955071649153, 13.88332582616279, 15.19485946813402, 15.54627413017915, 15.98137521703121, 16.37127098002175, 17.22276078102789
