| L(s) = 1 | + (−1.41 + 1.73i)5-s − 1.41i·7-s − 2i·11-s − 5.65·13-s + 4.89i·17-s + 6i·19-s − 7.07i·23-s + (−0.999 − 4.89i)25-s − 6.92i·29-s + 6.92·31-s + (2.44 + 2.00i)35-s + 2.82·37-s + 4·41-s + 2.44·43-s + 4.24i·47-s + ⋯ |
| L(s) = 1 | + (−0.632 + 0.774i)5-s − 0.534i·7-s − 0.603i·11-s − 1.56·13-s + 1.18i·17-s + 1.37i·19-s − 1.47i·23-s + (−0.199 − 0.979i)25-s − 1.28i·29-s + 1.24·31-s + (0.414 + 0.338i)35-s + 0.464·37-s + 0.624·41-s + 0.373·43-s + 0.618i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 + 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.223113815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.223113815\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.41 - 1.73i)T \) |
| good | 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 + 7.07iT - 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 6.92T + 31T^{2} \) |
| 37 | \( 1 - 2.82T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 2.44T + 43T^{2} \) |
| 47 | \( 1 - 4.24iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 2iT - 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 2.44T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 + 4.89iT - 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 14.6iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.430389527155872170728454440438, −7.946529441741827955912547230236, −7.33666051950574115253064750146, −6.39756536465634955884422143796, −5.90638823594775052045831240817, −4.52388401196544883484405029763, −4.07126843380706656396347160438, −3.03160975631890653454451251441, −2.19340794257297939592616934722, −0.54476751770788930972896517250,
0.810427589559486585421597239344, 2.25809177002569299987594503091, 3.05556440908115048893311973353, 4.31040657548637097609423467049, 5.01109819482624012167088668702, 5.38223090216390742845576621963, 6.81558238288248311473350199253, 7.35392260139187881693659940397, 7.955299134860919400173407916889, 9.150428387531936724278905809002
