| L(s) = 1 | + 3-s + 1.55i·5-s − 2.96i·7-s + 9-s + 2.24i·11-s + 1.55i·15-s + 1.01·17-s − 5.35i·19-s − 2.96i·21-s − 0.782·23-s + 2.58·25-s + 27-s + 2.69·29-s + 7.28i·31-s + 2.24i·33-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.695i·5-s − 1.11i·7-s + 0.333·9-s + 0.677i·11-s + 0.401i·15-s + 0.247·17-s − 1.22i·19-s − 0.646i·21-s − 0.163·23-s + 0.516·25-s + 0.192·27-s + 0.499·29-s + 1.30i·31-s + 0.391i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.427573127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.427573127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 1.55iT - 5T^{2} \) |
| 7 | \( 1 + 2.96iT - 7T^{2} \) |
| 11 | \( 1 - 2.24iT - 11T^{2} \) |
| 17 | \( 1 - 1.01T + 17T^{2} \) |
| 19 | \( 1 + 5.35iT - 19T^{2} \) |
| 23 | \( 1 + 0.782T + 23T^{2} \) |
| 29 | \( 1 - 2.69T + 29T^{2} \) |
| 31 | \( 1 - 7.28iT - 31T^{2} \) |
| 37 | \( 1 + 7.80iT - 37T^{2} \) |
| 41 | \( 1 - 5.34iT - 41T^{2} \) |
| 43 | \( 1 + 3.12T + 43T^{2} \) |
| 47 | \( 1 + 7.13iT - 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 4.35iT - 59T^{2} \) |
| 61 | \( 1 - 5.14T + 61T^{2} \) |
| 67 | \( 1 + 12.9iT - 67T^{2} \) |
| 71 | \( 1 - 13.8iT - 71T^{2} \) |
| 73 | \( 1 + 8.30iT - 73T^{2} \) |
| 79 | \( 1 - 1.23T + 79T^{2} \) |
| 83 | \( 1 - 7.67iT - 83T^{2} \) |
| 89 | \( 1 - 0.494iT - 89T^{2} \) |
| 97 | \( 1 + 11.1iT - 97T^{2} \) |
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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
−8.395924525386119753944049070795, −7.48951742359868268444981251006, −6.99813633507808239035943719934, −6.60223557193527316784386876645, −5.27715505939220268517488940921, −4.51652480500288157007431656865, −3.71922438506340871647904369108, −2.95857097636597015222683276335, −2.05459677083953340158483674486, −0.794229663308374786265739739641,
0.990151024950199176569300151902, 2.09277952694145078016116976054, 2.93375583494545520557730594951, 3.82157062651297033718174301426, 4.70228220137091099969415294837, 5.63074145731740150627549500255, 6.02131323710105429164857029114, 7.10368064401083590360642479804, 8.055947155954603904058065786793, 8.489824756207978056397458360504
