L(s) = 1 | − 2-s + 2·3-s + 4-s + 2·5-s − 2·6-s + 2·7-s − 8-s + 9-s − 2·10-s − 11-s + 2·12-s − 2·13-s − 2·14-s + 4·15-s + 16-s − 6·17-s − 18-s + 2·20-s + 4·21-s + 22-s − 2·24-s − 25-s + 2·26-s − 4·27-s + 2·28-s − 2·29-s − 4·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.894·5-s − 0.816·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s + 0.577·12-s − 0.554·13-s − 0.534·14-s + 1.03·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.447·20-s + 0.872·21-s + 0.213·22-s − 0.408·24-s − 1/5·25-s + 0.392·26-s − 0.769·27-s + 0.377·28-s − 0.371·29-s − 0.730·30-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 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.69938598332186259867216934677, −7.02553326906149803508105968685, −6.27069307967945204937986403584, −5.40949627748415142883247623962, −4.72024103200300233436214624959, −3.68951187254623807673954852442, −2.79745035634974018820852356212, −2.02878714193227454534285498009, −1.72536263148644519711541882318, 0,
1.72536263148644519711541882318, 2.02878714193227454534285498009, 2.79745035634974018820852356212, 3.68951187254623807673954852442, 4.72024103200300233436214624959, 5.40949627748415142883247623962, 6.27069307967945204937986403584, 7.02553326906149803508105968685, 7.69938598332186259867216934677