| L(s) = 1 | + (0.707 − 0.707i)5-s + (−0.471 − 1.76i)7-s + (0.870 − 3.24i)11-s + (3.33 − 1.36i)13-s + (−0.219 + 0.380i)17-s + (2.10 − 0.565i)19-s + (2.61 + 4.52i)23-s − 1.00i·25-s + (1.00 − 0.582i)29-s + (1.90 + 1.90i)31-s + (−1.57 − 0.911i)35-s + (−4.97 − 1.33i)37-s + (−10.9 − 2.92i)41-s + (−2.13 − 1.23i)43-s + (0.175 + 0.175i)47-s + ⋯ |
| L(s) = 1 | + (0.316 − 0.316i)5-s + (−0.178 − 0.665i)7-s + (0.262 − 0.979i)11-s + (0.925 − 0.377i)13-s + (−0.0533 + 0.0923i)17-s + (0.483 − 0.129i)19-s + (0.545 + 0.944i)23-s − 0.200i·25-s + (0.187 − 0.108i)29-s + (0.341 + 0.341i)31-s + (−0.266 − 0.154i)35-s + (−0.818 − 0.219i)37-s + (−1.70 − 0.457i)41-s + (−0.326 − 0.188i)43-s + (0.0256 + 0.0256i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0723 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0723 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.821365920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.821365920\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 + 0.707i)T \) |
| 13 | \( 1 + (-3.33 + 1.36i)T \) |
| good | 7 | \( 1 + (0.471 + 1.76i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.870 + 3.24i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (0.219 - 0.380i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.10 + 0.565i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-2.61 - 4.52i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.00 + 0.582i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.90 - 1.90i)T + 31iT^{2} \) |
| 37 | \( 1 + (4.97 + 1.33i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (10.9 + 2.92i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (2.13 + 1.23i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.175 - 0.175i)T + 47iT^{2} \) |
| 53 | \( 1 + 10.6iT - 53T^{2} \) |
| 59 | \( 1 + (7.48 - 2.00i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-5.54 + 9.60i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.16 - 4.35i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (0.928 + 3.46i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-7.33 + 7.33i)T - 73iT^{2} \) |
| 79 | \( 1 - 5.25T + 79T^{2} \) |
| 83 | \( 1 + (8.32 - 8.32i)T - 83iT^{2} \) |
| 89 | \( 1 + (1.69 - 6.34i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-18.2 + 4.89i)T + (84.0 - 48.5i)T^{2} \) |
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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
−8.688732700805479679581055615207, −8.239299348350033252761485416531, −7.17965689623662591314202814857, −6.49726349614953495122072896194, −5.63343205271188202942486383114, −4.95462474404314493586744020110, −3.68087726793889162289209387880, −3.26661801353782144808344001867, −1.68912484055960553144740190503, −0.66391073447278906894014194589,
1.38206130664991703042015926891, 2.41671459580741797036928710033, 3.34770423571465404621664161634, 4.41369460939654776338291738365, 5.23536679945063008068923119921, 6.20483150534836012385904656960, 6.72313487762735932921167810847, 7.56545013329765207202131796985, 8.618233160376897726927538116926, 9.065008731796833598270510871825
