| L(s) = 1 | + (2 + i)5-s − 2i·7-s + 2·11-s − 6i·13-s − 2i·17-s + 4i·23-s + (3 + 4i)25-s + 8·31-s + (2 − 4i)35-s + 2i·37-s − 2·41-s − 4i·43-s − 8i·47-s + 3·49-s + 6i·53-s + ⋯ |
| L(s) = 1 | + (0.894 + 0.447i)5-s − 0.755i·7-s + 0.603·11-s − 1.66i·13-s − 0.485i·17-s + 0.834i·23-s + (0.600 + 0.800i)25-s + 1.43·31-s + (0.338 − 0.676i)35-s + 0.328i·37-s − 0.312·41-s − 0.609i·43-s − 1.16i·47-s + 0.428·49-s + 0.824i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.76850 - 0.417486i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.76850 - 0.417486i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2 - i)T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 10T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−10.22913772100440830577826844760, −9.764343617008022671293880492594, −8.662422783980672968610042698474, −7.63705598331460257159777406841, −6.84168696337740343179673921397, −5.90010035099999057095378839903, −5.03860714797579347515688127826, −3.67566400256894938934555642636, −2.67488333729237000131898752914, −1.09800954351398469309793518454,
1.53278371253038712821949014209, 2.55175790639803455243570848933, 4.16511323275093133792461951340, 5.01943500752975389755325648208, 6.24744767974066892109443720473, 6.57566810497671810407901358358, 8.094475984055633904972492274591, 9.001869391244456565021831771862, 9.384141790481265150351677797226, 10.35185190618789454496172302198
