L(s) = 1 | + 3-s + 5-s + 4.81·7-s + 9-s + 1.45·11-s − 2.81·13-s + 15-s + 6.26·17-s − 19-s + 4.81·21-s − 6.26·23-s + 25-s + 27-s + 0.548·29-s − 8.26·31-s + 1.45·33-s + 4.81·35-s + 6.81·37-s − 2.81·39-s + 4.54·41-s + 7.71·43-s + 45-s + 10.2·47-s + 16.1·49-s + 6.26·51-s − 0.265·53-s + 1.45·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.81·7-s + 0.333·9-s + 0.437·11-s − 0.780·13-s + 0.258·15-s + 1.51·17-s − 0.229·19-s + 1.05·21-s − 1.30·23-s + 0.200·25-s + 0.192·27-s + 0.101·29-s − 1.48·31-s + 0.252·33-s + 0.813·35-s + 1.12·37-s − 0.450·39-s + 0.710·41-s + 1.17·43-s + 0.149·45-s + 1.49·47-s + 2.30·49-s + 0.877·51-s − 0.0364·53-s + 0.195·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.560258038\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.560258038\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 4.81T + 7T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 + 2.81T + 13T^{2} \) |
| 17 | \( 1 - 6.26T + 17T^{2} \) |
| 23 | \( 1 + 6.26T + 23T^{2} \) |
| 29 | \( 1 - 0.548T + 29T^{2} \) |
| 31 | \( 1 + 8.26T + 31T^{2} \) |
| 37 | \( 1 - 6.81T + 37T^{2} \) |
| 41 | \( 1 - 4.54T + 41T^{2} \) |
| 43 | \( 1 - 7.71T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 0.265T + 53T^{2} \) |
| 59 | \( 1 + 2.90T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 6.90T + 71T^{2} \) |
| 73 | \( 1 + 6.53T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 9.16T + 83T^{2} \) |
| 89 | \( 1 + 18.7T + 89T^{2} \) |
| 97 | \( 1 + 6.81T + 97T^{2} \) |
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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
−8.296799481418709519383355564871, −7.54088359570004064873408694631, −7.31947812478263028776705568286, −5.85862737715754439959853550365, −5.50078450827291010944317078384, −4.46436373347290706958299627214, −3.96572390799856246632232345208, −2.68241167661707197994177492881, −1.93659036864651826985359685734, −1.13126660772651465550191689324,
1.13126660772651465550191689324, 1.93659036864651826985359685734, 2.68241167661707197994177492881, 3.96572390799856246632232345208, 4.46436373347290706958299627214, 5.50078450827291010944317078384, 5.85862737715754439959853550365, 7.31947812478263028776705568286, 7.54088359570004064873408694631, 8.296799481418709519383355564871