L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s + 2·7-s − 8-s + 9-s + 11-s + 2·12-s + 13-s − 2·14-s + 16-s − 6·17-s − 18-s + 4·21-s − 22-s − 2·24-s − 5·25-s − 26-s − 4·27-s + 2·28-s − 9·29-s − 8·31-s − 32-s + 2·33-s + 6·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.577·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.872·21-s − 0.213·22-s − 0.408·24-s − 25-s − 0.196·26-s − 0.769·27-s + 0.377·28-s − 1.67·29-s − 1.43·31-s − 0.176·32-s + 0.348·33-s + 1.02·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−7.64395224406250001767139461291, −7.22258910335469474746435651282, −6.15086401116504048458423564659, −5.59874993569980893690782509603, −4.36703334567352177321557879240, −3.88430112142237559628316879448, −2.87676187333491317227927523670, −2.10467573190885388754236000308, −1.54892809830414465965986935841, 0,
1.54892809830414465965986935841, 2.10467573190885388754236000308, 2.87676187333491317227927523670, 3.88430112142237559628316879448, 4.36703334567352177321557879240, 5.59874993569980893690782509603, 6.15086401116504048458423564659, 7.22258910335469474746435651282, 7.64395224406250001767139461291