| L(s) = 1 | − 2.82i·3-s − 5.00·9-s + 4·11-s + 4.24i·13-s − 1.41i·17-s + 2.82·19-s + 4i·23-s + 5.65i·27-s − 8·29-s − 11.3i·33-s − 8i·37-s + 12·39-s + 7.07·41-s + 4i·43-s − 5.65i·47-s + ⋯ |
| L(s) = 1 | − 1.63i·3-s − 1.66·9-s + 1.20·11-s + 1.17i·13-s − 0.342i·17-s + 0.648·19-s + 0.834i·23-s + 1.08i·27-s − 1.48·29-s − 1.96i·33-s − 1.31i·37-s + 1.92·39-s + 1.10·41-s + 0.609i·43-s − 0.825i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.927413266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.927413266\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2.82iT - 3T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 + 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 7.07T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 5.65iT - 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 7.07T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 7.07iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 14.1iT - 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 - 1.41iT - 97T^{2} \) |
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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
−7.76140870102857892583229881434, −7.21327589518969630366790920138, −6.80151996840542812753454052901, −6.02729378527902563829175172777, −5.38291065844333338900086532699, −4.16401024298539552654884941928, −3.41346666987023794485825046778, −2.15533590537144192532002639065, −1.65750487900442208929034329180, −0.62964064971181029517792810010,
0.995340850676706747805865541988, 2.53020339758193569706609674908, 3.45982731645205563749420727763, 3.95648307281840728643775085708, 4.71309147701084800728177071472, 5.50672533500317408973528878005, 6.04681689062289354129969288886, 7.05143446670980353005974901782, 7.958529348284923299817939255830, 8.726273112103482147210621405008
