L(s) = 1 | − 1.61·2-s + 3-s + 0.618·4-s − 1.61·6-s + 0.618·7-s + 2.23·8-s − 2·9-s − 5.23·11-s + 0.618·12-s + 1.85·13-s − 1.00·14-s − 4.85·16-s − 5.23·17-s + 3.23·18-s + 0.854·19-s + 0.618·21-s + 8.47·22-s + 3.76·23-s + 2.23·24-s − 3·26-s − 5·27-s + 0.381·28-s − 3.61·29-s − 3·31-s + 3.38·32-s − 5.23·33-s + 8.47·34-s + ⋯ |
L(s) = 1 | − 1.14·2-s + 0.577·3-s + 0.309·4-s − 0.660·6-s + 0.233·7-s + 0.790·8-s − 0.666·9-s − 1.57·11-s + 0.178·12-s + 0.514·13-s − 0.267·14-s − 1.21·16-s − 1.26·17-s + 0.762·18-s + 0.195·19-s + 0.134·21-s + 1.80·22-s + 0.784·23-s + 0.456·24-s − 0.588·26-s − 0.962·27-s + 0.0721·28-s − 0.671·29-s − 0.538·31-s + 0.597·32-s − 0.911·33-s + 1.45·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 1.85T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 - 0.854T + 19T^{2} \) |
| 23 | \( 1 - 3.76T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 0.236T + 37T^{2} \) |
| 41 | \( 1 + 0.763T + 41T^{2} \) |
| 43 | \( 1 + 4.85T + 43T^{2} \) |
| 47 | \( 1 - 0.618T + 47T^{2} \) |
| 53 | \( 1 + 3.47T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 8.70T + 61T^{2} \) |
| 67 | \( 1 - 4.76T + 67T^{2} \) |
| 71 | \( 1 + 6.61T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 + 8.09T + 79T^{2} \) |
| 83 | \( 1 + 6.23T + 83T^{2} \) |
| 89 | \( 1 + 8.94T + 89T^{2} \) |
| 97 | \( 1 + 3.85T + 97T^{2} \) |
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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
−10.01652423904056671200617946383, −9.101642758873869954633728325187, −8.506646481105811914530586321941, −7.88479164562301327349912281171, −7.01893864734912219439462109009, −5.60320073730555610124605744204, −4.58034193435069674182828980796, −3.07879746442872269965969755136, −1.92322418237009403501123881674, 0,
1.92322418237009403501123881674, 3.07879746442872269965969755136, 4.58034193435069674182828980796, 5.60320073730555610124605744204, 7.01893864734912219439462109009, 7.88479164562301327349912281171, 8.506646481105811914530586321941, 9.101642758873869954633728325187, 10.01652423904056671200617946383