| L(s) = 1 | − 2.30i·5-s − 4.59i·7-s − 3.79·11-s − 2.59·13-s + 3.13i·17-s − i·19-s + 3.59·23-s − 0.333·25-s − 7.15i·29-s − 6.83i·31-s − 10.6·35-s − 5.26·37-s + 10.9i·41-s − 5.05i·43-s + 2.74·47-s + ⋯ |
| L(s) = 1 | − 1.03i·5-s − 1.73i·7-s − 1.14·11-s − 0.719·13-s + 0.761i·17-s − 0.229i·19-s + 0.750·23-s − 0.0666·25-s − 1.32i·29-s − 1.22i·31-s − 1.79·35-s − 0.864·37-s + 1.71i·41-s − 0.771i·43-s + 0.400·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7108077161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7108077161\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 + 2.30iT - 5T^{2} \) |
| 7 | \( 1 + 4.59iT - 7T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 + 2.59T + 13T^{2} \) |
| 17 | \( 1 - 3.13iT - 17T^{2} \) |
| 23 | \( 1 - 3.59T + 23T^{2} \) |
| 29 | \( 1 + 7.15iT - 29T^{2} \) |
| 31 | \( 1 + 6.83iT - 31T^{2} \) |
| 37 | \( 1 + 5.26T + 37T^{2} \) |
| 41 | \( 1 - 10.9iT - 41T^{2} \) |
| 43 | \( 1 + 5.05iT - 43T^{2} \) |
| 47 | \( 1 - 2.74T + 47T^{2} \) |
| 53 | \( 1 - 0.137iT - 53T^{2} \) |
| 59 | \( 1 + 3.11T + 59T^{2} \) |
| 61 | \( 1 + 1.56T + 61T^{2} \) |
| 67 | \( 1 - 2.53iT - 67T^{2} \) |
| 71 | \( 1 + 4.05T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 11.5iT - 79T^{2} \) |
| 83 | \( 1 - 7.81T + 83T^{2} \) |
| 89 | \( 1 - 6.79iT - 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.203965566798715427704197493130, −7.72265138612458652161705263467, −7.06349674028485914406545872019, −6.06073234017182229705744294765, −5.03508285351411864512402072714, −4.52965206838989196858735749277, −3.74237181723847465385294811692, −2.52586062510770099970894512682, −1.21185952821603120024399430978, −0.23424487555399625421027024879,
1.94794173051809876866867585871, 2.85591510485940399136932214858, 3.15931247349540774059351384282, 4.93667814477818082912402832490, 5.27614983729960616268907114022, 6.17486254463738451973550028525, 7.08523992589234713161792849716, 7.56520196936671314587071291760, 8.763526397327386711193355361150, 8.992438274447017358540156783792
