L(s) = 1 | + 2-s − 4-s + 4.47·7-s − 3·8-s + 1.23·11-s − 5.61·13-s + 4.47·14-s − 16-s + 3.85·17-s + 1.23·19-s + 1.23·22-s + 4.47·23-s − 5.61·26-s − 4.47·28-s − 6.61·29-s + 2.76·31-s + 5·32-s + 3.85·34-s + 3.09·37-s + 1.23·38-s + 3.61·41-s − 7.70·43-s − 1.23·44-s + 4.47·46-s − 0.763·47-s + 13.0·49-s + 5.61·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.5·4-s + 1.69·7-s − 1.06·8-s + 0.372·11-s − 1.55·13-s + 1.19·14-s − 0.250·16-s + 0.934·17-s + 0.283·19-s + 0.263·22-s + 0.932·23-s − 1.10·26-s − 0.845·28-s − 1.22·29-s + 0.496·31-s + 0.883·32-s + 0.660·34-s + 0.508·37-s + 0.200·38-s + 0.565·41-s − 1.17·43-s − 0.186·44-s + 0.659·46-s − 0.111·47-s + 1.85·49-s + 0.779·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.774878795\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.774878795\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 - T + 2T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 + 5.61T + 13T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 19 | \( 1 - 1.23T + 19T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 6.61T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 - 3.09T + 37T^{2} \) |
| 41 | \( 1 - 3.61T + 41T^{2} \) |
| 43 | \( 1 + 7.70T + 43T^{2} \) |
| 47 | \( 1 + 0.763T + 47T^{2} \) |
| 53 | \( 1 - 3.61T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 1.61T + 61T^{2} \) |
| 67 | \( 1 + 0.763T + 67T^{2} \) |
| 71 | \( 1 - 5.23T + 71T^{2} \) |
| 73 | \( 1 - 8.09T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 5.38T + 89T^{2} \) |
| 97 | \( 1 + 2.14T + 97T^{2} \) |
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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
−8.001792396772810485823347346665, −7.55980602276251211322644190509, −6.70137965771809788979893282411, −5.54358709426977098850322106047, −5.18295119847733815065438716795, −4.62885844592880637460708919260, −3.88282932023147340642541278734, −2.92712935824126354231928550165, −1.96273664905126820344905425208, −0.823830784869458668129635927313,
0.823830784869458668129635927313, 1.96273664905126820344905425208, 2.92712935824126354231928550165, 3.88282932023147340642541278734, 4.62885844592880637460708919260, 5.18295119847733815065438716795, 5.54358709426977098850322106047, 6.70137965771809788979893282411, 7.55980602276251211322644190509, 8.001792396772810485823347346665