| L(s) = 1 | + 2-s − 2.61·3-s + 4-s − 2.85·5-s − 2.61·6-s + 7-s + 8-s + 3.85·9-s − 2.85·10-s + 4.61·11-s − 2.61·12-s − 5.23·13-s + 14-s + 7.47·15-s + 16-s + 2.47·17-s + 3.85·18-s − 2.85·20-s − 2.61·21-s + 4.61·22-s − 5.70·23-s − 2.61·24-s + 3.14·25-s − 5.23·26-s − 2.23·27-s + 28-s + 8.85·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.51·3-s + 0.5·4-s − 1.27·5-s − 1.06·6-s + 0.377·7-s + 0.353·8-s + 1.28·9-s − 0.902·10-s + 1.39·11-s − 0.755·12-s − 1.45·13-s + 0.267·14-s + 1.92·15-s + 0.250·16-s + 0.599·17-s + 0.908·18-s − 0.638·20-s − 0.571·21-s + 0.984·22-s − 1.19·23-s − 0.534·24-s + 0.629·25-s − 1.02·26-s − 0.430·27-s + 0.188·28-s + 1.64·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.155610680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.155610680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 11 | \( 1 - 4.61T + 11T^{2} \) |
| 13 | \( 1 + 5.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 - 8.85T + 29T^{2} \) |
| 31 | \( 1 - 0.472T + 31T^{2} \) |
| 37 | \( 1 + 4.85T + 37T^{2} \) |
| 41 | \( 1 + 5.32T + 41T^{2} \) |
| 43 | \( 1 + 7.61T + 43T^{2} \) |
| 47 | \( 1 + 3.85T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 8.85T + 59T^{2} \) |
| 61 | \( 1 + 2.32T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + 7.23T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 8.94T + 83T^{2} \) |
| 89 | \( 1 - 1.38T + 89T^{2} \) |
| 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
−7.990484800497889240384824630941, −7.21433769115620377129407222380, −6.74566587582782840978527525201, −6.02205332902358180495311574204, −5.16943988569007445921109112002, −4.60081715973733633038865698681, −4.06075393716654564195864050623, −3.16217422709710157698057640389, −1.74505526392658759722624175652, −0.57483547744143112307305337657,
0.57483547744143112307305337657, 1.74505526392658759722624175652, 3.16217422709710157698057640389, 4.06075393716654564195864050623, 4.60081715973733633038865698681, 5.16943988569007445921109112002, 6.02205332902358180495311574204, 6.74566587582782840978527525201, 7.21433769115620377129407222380, 7.990484800497889240384824630941
