L(s) = 1 | − 2.61·2-s − 3-s + 4.85·4-s − 2.23·5-s + 2.61·6-s − 7.47·8-s + 9-s + 5.85·10-s − 11-s − 4.85·12-s + 3.47·13-s + 2.23·15-s + 9.85·16-s + 6·17-s − 2.61·18-s − 4.23·19-s − 10.8·20-s + 2.61·22-s − 2.47·23-s + 7.47·24-s − 9.09·26-s − 27-s − 10.2·29-s − 5.85·30-s + 8.47·31-s − 10.8·32-s + 33-s + ⋯ |
L(s) = 1 | − 1.85·2-s − 0.577·3-s + 2.42·4-s − 0.999·5-s + 1.06·6-s − 2.64·8-s + 0.333·9-s + 1.85·10-s − 0.301·11-s − 1.40·12-s + 0.962·13-s + 0.577·15-s + 2.46·16-s + 1.45·17-s − 0.617·18-s − 0.971·19-s − 2.42·20-s + 0.558·22-s − 0.515·23-s + 1.52·24-s − 1.78·26-s − 0.192·27-s − 1.90·29-s − 1.06·30-s + 1.52·31-s − 1.91·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3705266302\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3705266302\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4.23T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 8.47T + 31T^{2} \) |
| 37 | \( 1 + 0.527T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 0.472T + 43T^{2} \) |
| 47 | \( 1 - 11.4T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 + 6.94T + 61T^{2} \) |
| 67 | \( 1 - 0.708T + 67T^{2} \) |
| 71 | \( 1 - 4.47T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 2.47T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 6.94T + 97T^{2} \) |
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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
−9.400317977913097710570593126020, −8.524649432781679522984719247707, −7.79411292934807206931554205359, −7.50606998763773376492684119385, −6.34905405829755616043889290647, −5.77257477189267214362641864231, −4.22868064781601819288147691951, −3.16495715504391030713362267660, −1.74450108306809813732576957408, −0.58023986852057187785248996683,
0.58023986852057187785248996683, 1.74450108306809813732576957408, 3.16495715504391030713362267660, 4.22868064781601819288147691951, 5.77257477189267214362641864231, 6.34905405829755616043889290647, 7.50606998763773376492684119385, 7.79411292934807206931554205359, 8.524649432781679522984719247707, 9.400317977913097710570593126020