| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 3·7-s + 8-s + 9-s + 3·11-s + 12-s + 2·13-s − 3·14-s + 16-s − 8·17-s + 18-s − 7·19-s − 3·21-s + 3·22-s − 7·23-s + 24-s + 2·26-s + 27-s − 3·28-s − 8·29-s − 31-s + 32-s + 3·33-s − 8·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 0.554·13-s − 0.801·14-s + 1/4·16-s − 1.94·17-s + 0.235·18-s − 1.60·19-s − 0.654·21-s + 0.639·22-s − 1.45·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s − 0.566·28-s − 1.48·29-s − 0.179·31-s + 0.176·32-s + 0.522·33-s − 1.37·34-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 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.964962895542803632321234472505, −6.95070235448631663386120436832, −6.34687305291337051056450608058, −6.10929687586275947608770222667, −4.73926230768719502207188999764, −3.91882763570167253587417993328, −3.66339523024137065248312673295, −2.43171742572825760487443871273, −1.83126019783338753189132116274, 0,
1.83126019783338753189132116274, 2.43171742572825760487443871273, 3.66339523024137065248312673295, 3.91882763570167253587417993328, 4.73926230768719502207188999764, 6.10929687586275947608770222667, 6.34687305291337051056450608058, 6.95070235448631663386120436832, 7.964962895542803632321234472505
