L(s) = 1 | + 3-s + 7-s + 9-s − 2·11-s + 13-s − 3·17-s + 21-s − 23-s + 27-s − 5·29-s − 7·31-s − 2·33-s + 2·37-s + 39-s + 7·41-s − 11·43-s + 8·47-s + 49-s − 3·51-s + 53-s + 5·59-s − 3·61-s + 63-s − 12·67-s − 69-s − 12·71-s + 6·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.727·17-s + 0.218·21-s − 0.208·23-s + 0.192·27-s − 0.928·29-s − 1.25·31-s − 0.348·33-s + 0.328·37-s + 0.160·39-s + 1.09·41-s − 1.67·43-s + 1.16·47-s + 1/7·49-s − 0.420·51-s + 0.137·53-s + 0.650·59-s − 0.384·61-s + 0.125·63-s − 1.46·67-s − 0.120·69-s − 1.42·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.46975734554011768308968606120, −6.98988028840389158587917507818, −5.97864255419351776753845638556, −5.41707373121940344472610415336, −4.50609001407409696358126615863, −3.91767014078316481521425736331, −3.02186978729451114059720153102, −2.23233777379663727508323601325, −1.44651527142140040108571135130, 0,
1.44651527142140040108571135130, 2.23233777379663727508323601325, 3.02186978729451114059720153102, 3.91767014078316481521425736331, 4.50609001407409696358126615863, 5.41707373121940344472610415336, 5.97864255419351776753845638556, 6.98988028840389158587917507818, 7.46975734554011768308968606120