| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s − 3·11-s − 12-s − 4·13-s + 14-s + 16-s + 18-s − 19-s − 21-s − 3·22-s + 5·23-s − 24-s − 4·26-s − 27-s + 28-s + 2·29-s − 31-s + 32-s + 3·33-s + 36-s + 4·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.235·18-s − 0.229·19-s − 0.218·21-s − 0.639·22-s + 1.04·23-s − 0.204·24-s − 0.784·26-s − 0.192·27-s + 0.188·28-s + 0.371·29-s − 0.179·31-s + 0.176·32-s + 0.522·33-s + 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 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
−7.79256174634508703215331824821, −7.07651993949604817375911864266, −6.48261260676149534898680174843, −5.51949176213551209038009475863, −4.96241793973118221790469590808, −4.52870010329789945915439578897, −3.32479751452340829703395613488, −2.54609490410073069086706085470, −1.51477846087782443381191706736, 0,
1.51477846087782443381191706736, 2.54609490410073069086706085470, 3.32479751452340829703395613488, 4.52870010329789945915439578897, 4.96241793973118221790469590808, 5.51949176213551209038009475863, 6.48261260676149534898680174843, 7.07651993949604817375911864266, 7.79256174634508703215331824821
