| L(s) = 1 | + 1.85·3-s − 1.80·5-s + 0.458·9-s + 5.99·11-s − 4.33·13-s − 3.36·15-s + 7.43·17-s − 2.28·19-s + 3.07·23-s − 1.73·25-s − 4.72·27-s + 5.91·29-s − 3.61·31-s + 11.1·33-s + 8.67·37-s − 8.06·39-s + 11.6·41-s − 4.17·43-s − 0.829·45-s − 2.97·47-s + 13.8·51-s − 8.17·53-s − 10.8·55-s − 4.25·57-s + 13.5·59-s − 1.99·61-s + 7.83·65-s + ⋯ |
| L(s) = 1 | + 1.07·3-s − 0.808·5-s + 0.152·9-s + 1.80·11-s − 1.20·13-s − 0.867·15-s + 1.80·17-s − 0.524·19-s + 0.640·23-s − 0.346·25-s − 0.909·27-s + 1.09·29-s − 0.648·31-s + 1.94·33-s + 1.42·37-s − 1.29·39-s + 1.82·41-s − 0.636·43-s − 0.123·45-s − 0.434·47-s + 1.93·51-s − 1.12·53-s − 1.46·55-s − 0.563·57-s + 1.76·59-s − 0.255·61-s + 0.972·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.463780252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.463780252\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.85T + 3T^{2} \) |
| 5 | \( 1 + 1.80T + 5T^{2} \) |
| 11 | \( 1 - 5.99T + 11T^{2} \) |
| 13 | \( 1 + 4.33T + 13T^{2} \) |
| 17 | \( 1 - 7.43T + 17T^{2} \) |
| 19 | \( 1 + 2.28T + 19T^{2} \) |
| 23 | \( 1 - 3.07T + 23T^{2} \) |
| 29 | \( 1 - 5.91T + 29T^{2} \) |
| 31 | \( 1 + 3.61T + 31T^{2} \) |
| 37 | \( 1 - 8.67T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 4.17T + 43T^{2} \) |
| 47 | \( 1 + 2.97T + 47T^{2} \) |
| 53 | \( 1 + 8.17T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 1.99T + 61T^{2} \) |
| 67 | \( 1 - 7.69T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 6.60T + 73T^{2} \) |
| 79 | \( 1 + 0.873T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 1.70T + 89T^{2} \) |
| 97 | \( 1 - 7.23T + 97T^{2} \) |
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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
−8.754382533491359102475501009270, −8.036056024043727945108427164266, −7.54102922587105937305619152915, −6.73993710633311330407360531106, −5.78907122391676381574458918547, −4.65461332436788244733679036763, −3.84610606981074430997073689591, −3.25489602625604311180330513261, −2.27034282958770242087546560371, −0.960922098337462033098053333821,
0.960922098337462033098053333821, 2.27034282958770242087546560371, 3.25489602625604311180330513261, 3.84610606981074430997073689591, 4.65461332436788244733679036763, 5.78907122391676381574458918547, 6.73993710633311330407360531106, 7.54102922587105937305619152915, 8.036056024043727945108427164266, 8.754382533491359102475501009270
