| L(s) = 1 | + (−0.839 + 1.51i)3-s + i·5-s + i·7-s + (−1.59 − 2.54i)9-s + (−1.39 − 3.01i)11-s + 1.12i·13-s + (−1.51 − 0.839i)15-s + 6.62·17-s − 6.33i·19-s + (−1.51 − 0.839i)21-s + 6.80i·23-s − 25-s + (5.18 − 0.276i)27-s − 0.173·29-s − 7.28·31-s + ⋯ |
| L(s) = 1 | + (−0.484 + 0.874i)3-s + 0.447i·5-s + 0.377i·7-s + (−0.530 − 0.847i)9-s + (−0.419 − 0.907i)11-s + 0.311i·13-s + (−0.391 − 0.216i)15-s + 1.60·17-s − 1.45i·19-s + (−0.330 − 0.183i)21-s + 1.41i·23-s − 0.200·25-s + (0.998 − 0.0532i)27-s − 0.0321·29-s − 1.30·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7497887284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7497887284\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.839 - 1.51i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (1.39 + 3.01i)T \) |
| good | 13 | \( 1 - 1.12iT - 13T^{2} \) |
| 17 | \( 1 - 6.62T + 17T^{2} \) |
| 19 | \( 1 + 6.33iT - 19T^{2} \) |
| 23 | \( 1 - 6.80iT - 23T^{2} \) |
| 29 | \( 1 + 0.173T + 29T^{2} \) |
| 31 | \( 1 + 7.28T + 31T^{2} \) |
| 37 | \( 1 - 5.43T + 37T^{2} \) |
| 41 | \( 1 + 6.61T + 41T^{2} \) |
| 43 | \( 1 - 8.97iT - 43T^{2} \) |
| 47 | \( 1 - 3.13iT - 47T^{2} \) |
| 53 | \( 1 - 7.05iT - 53T^{2} \) |
| 59 | \( 1 + 0.806iT - 59T^{2} \) |
| 61 | \( 1 - 11.3iT - 61T^{2} \) |
| 67 | \( 1 - 0.188T + 67T^{2} \) |
| 71 | \( 1 + 11.7iT - 71T^{2} \) |
| 73 | \( 1 - 7.05iT - 73T^{2} \) |
| 79 | \( 1 - 3.47iT - 79T^{2} \) |
| 83 | \( 1 + 6.82T + 83T^{2} \) |
| 89 | \( 1 + 0.0346iT - 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
| 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
−8.841320870597490398090811795417, −7.933487289365205105822684766308, −7.23222957914488279291704813380, −6.28737398110533965168365473960, −5.60867887243402773363480340285, −5.18767497559766787391840804229, −4.15168305033022140212071150299, −3.27229940921081832163744875947, −2.79034283030871742273029826167, −1.19170319216994309398543693688,
0.24045857993686204796780208348, 1.38189666244840521768360086880, 2.15328107177506988449143580613, 3.33528115118795884318692250832, 4.27685590501466443315589521962, 5.31909593383729554467778503748, 5.58494792012091456663717675864, 6.62476957729653131040035199504, 7.24396267812800879669182860894, 8.031059417059633842675230712330
