| L(s) = 1 | + 4i·7-s + 3·9-s + 4·11-s − 2i·13-s − 2i·17-s − 4·19-s + 4i·23-s + 2·29-s − 8·31-s − 6i·37-s − 6·41-s − 8i·43-s − 4i·47-s − 9·49-s + 6i·53-s + ⋯ |
| L(s) = 1 | + 1.51i·7-s + 9-s + 1.20·11-s − 0.554i·13-s − 0.485i·17-s − 0.917·19-s + 0.834i·23-s + 0.371·29-s − 1.43·31-s − 0.986i·37-s − 0.937·41-s − 1.21i·43-s − 0.583i·47-s − 1.28·49-s + 0.824i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24271 + 0.293364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24271 + 0.293364i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−12.38981497891100207714428544083, −11.79717199474207090526706509116, −10.57672707077590806904930820841, −9.370958191671415726732925723644, −8.787590035548931502752468186083, −7.40009193298089497516960432965, −6.26572895447990724875742180842, −5.16229961972730179546441189082, −3.69432978098095738914248953761, −1.97125972607415736782725038173,
1.45358696241604731067496319046, 3.84128333586622379389715092934, 4.49702032276493263404806241522, 6.51685402935686980154741072998, 7.05782374120953080129810321309, 8.342900122154824029657535891585, 9.594714414580979364379518245246, 10.41850683258951192488571265111, 11.29314486404453308023202215508, 12.52246034701159099877987113553
