| L(s) = 1 | + 2-s − 3-s + 4-s − 4.16·5-s − 6-s + 0.0852·7-s + 8-s + 9-s − 4.16·10-s + 1.68·11-s − 12-s + 13-s + 0.0852·14-s + 4.16·15-s + 16-s + 3.10·17-s + 18-s − 7.65·19-s − 4.16·20-s − 0.0852·21-s + 1.68·22-s + 3.94·23-s − 24-s + 12.3·25-s + 26-s − 27-s + 0.0852·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.86·5-s − 0.408·6-s + 0.0322·7-s + 0.353·8-s + 0.333·9-s − 1.31·10-s + 0.507·11-s − 0.288·12-s + 0.277·13-s + 0.0227·14-s + 1.07·15-s + 0.250·16-s + 0.752·17-s + 0.235·18-s − 1.75·19-s − 0.931·20-s − 0.0185·21-s + 0.358·22-s + 0.823·23-s − 0.204·24-s + 2.47·25-s + 0.196·26-s − 0.192·27-s + 0.0161·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.411665075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.411665075\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 4.16T + 5T^{2} \) |
| 7 | \( 1 - 0.0852T + 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 + 7.65T + 19T^{2} \) |
| 23 | \( 1 - 3.94T + 23T^{2} \) |
| 29 | \( 1 + 2.99T + 29T^{2} \) |
| 31 | \( 1 + 7.65T + 31T^{2} \) |
| 37 | \( 1 - 0.0664T + 37T^{2} \) |
| 41 | \( 1 + 5.31T + 41T^{2} \) |
| 43 | \( 1 - 2.96T + 43T^{2} \) |
| 47 | \( 1 + 7.78T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 0.189T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 9.00T + 71T^{2} \) |
| 73 | \( 1 - 2.51T + 73T^{2} \) |
| 79 | \( 1 + 8.51T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 0.620T + 89T^{2} \) |
| 97 | \( 1 - 3.38T + 97T^{2} \) |
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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
−7.67546622155310624645113063636, −6.98000408646943051074489677203, −6.61451737756891711944055945745, −5.60587591125746750318129734834, −4.95116466702170098351073486023, −4.10356905814256600607894086735, −3.83023714440835004085740938300, −3.02830964345499806264589260234, −1.73289100135665329720605039400, −0.54450959474699517871506938521,
0.54450959474699517871506938521, 1.73289100135665329720605039400, 3.02830964345499806264589260234, 3.83023714440835004085740938300, 4.10356905814256600607894086735, 4.95116466702170098351073486023, 5.60587591125746750318129734834, 6.61451737756891711944055945745, 6.98000408646943051074489677203, 7.67546622155310624645113063636
