L(s) = 1 | + (0.372 + 1.14i)2-s + (0.441 − 0.320i)4-s + (0.309 − 0.951i)5-s + (−3.49 + 2.53i)7-s + (2.48 + 1.80i)8-s + 1.20·10-s + (2.96 − 1.49i)11-s + (1.91 + 5.87i)13-s + (−4.20 − 3.05i)14-s + (−0.806 + 2.48i)16-s + (−0.248 + 0.764i)17-s + (1.51 + 1.10i)19-s + (−0.168 − 0.519i)20-s + (2.81 + 2.84i)22-s + 8.08·23-s + ⋯ |
L(s) = 1 | + (0.263 + 0.810i)2-s + (0.220 − 0.160i)4-s + (0.138 − 0.425i)5-s + (−1.31 + 0.958i)7-s + (0.878 + 0.637i)8-s + 0.381·10-s + (0.893 − 0.449i)11-s + (0.529 + 1.63i)13-s + (−1.12 − 0.817i)14-s + (−0.201 + 0.620i)16-s + (−0.0602 + 0.185i)17-s + (0.347 + 0.252i)19-s + (−0.0377 − 0.116i)20-s + (0.600 + 0.605i)22-s + 1.68·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.39926 + 1.13536i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.39926 + 1.13536i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 + (-2.96 + 1.49i)T \) |
good | 2 | \( 1 + (-0.372 - 1.14i)T + (-1.61 + 1.17i)T^{2} \) |
| 7 | \( 1 + (3.49 - 2.53i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-1.91 - 5.87i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (0.248 - 0.764i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.51 - 1.10i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 8.08T + 23T^{2} \) |
| 29 | \( 1 + (-0.857 + 0.622i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.499 + 1.53i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.48 + 3.25i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (4.08 + 2.96i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 9.87T + 43T^{2} \) |
| 47 | \( 1 + (10.5 + 7.63i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (0.823 + 2.53i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (5.34 - 3.88i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (1.65 - 5.08i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 7.49T + 67T^{2} \) |
| 71 | \( 1 + (1.80 - 5.54i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.88 - 2.82i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (4.31 + 13.2i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.72 + 11.4i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + (-0.416 - 1.28i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.40452633177147256346080986418, −10.06446771584735450978098700157, −9.090879291560768329903541260782, −8.669888134807646261250799800209, −7.10199428754260813779508471693, −6.43163408739860903700802188683, −5.84910297076287206584277474660, −4.68914954002362728902465427317, −3.32885318848866544365520580184, −1.73692072742484326813826045931,
1.13862353739453269143888065578, 3.13451845939131824923883872048, 3.32089613879751244545264599684, 4.74177142035892673209133460235, 6.39302418415157810300000990406, 6.92935642617613091991052392294, 7.88224643099921857729687029843, 9.386687067441628560869264890089, 10.09925282440076463401694493701, 10.76058782004880733774033730719