L(s) = 1 | + 38.1i·3-s + 102.·5-s − 512.·7-s − 726.·9-s − 2.41e3·11-s + 3.77e3i·13-s + 3.90e3i·15-s + 1.56e3·17-s + (2.73e3 + 6.28e3i)19-s − 1.95e4i·21-s + 626.·23-s − 5.12e3·25-s + 96.4i·27-s − 1.57e4i·29-s − 2.61e4i·31-s + ⋯ |
L(s) = 1 | + 1.41i·3-s + 0.819·5-s − 1.49·7-s − 0.996·9-s − 1.81·11-s + 1.71i·13-s + 1.15i·15-s + 0.318·17-s + (0.398 + 0.916i)19-s − 2.11i·21-s + 0.0514·23-s − 0.327·25-s + 0.00489i·27-s − 0.643i·29-s − 0.879i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.916i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.398 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.01445258899\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01445258899\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-2.73e3 - 6.28e3i)T \) |
good | 3 | \( 1 - 38.1iT - 729T^{2} \) |
| 5 | \( 1 - 102.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 512.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 2.41e3T + 1.77e6T^{2} \) |
| 13 | \( 1 - 3.77e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 1.56e3T + 2.41e7T^{2} \) |
| 23 | \( 1 - 626.T + 1.48e8T^{2} \) |
| 29 | \( 1 + 1.57e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 2.61e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 4.94e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 1.19e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.33e4T + 6.32e9T^{2} \) |
| 47 | \( 1 - 1.86e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 2.04e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.82e4iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.53e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 5.19e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.68e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 5.85e4T + 1.51e11T^{2} \) |
| 79 | \( 1 - 4.73e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 1.27e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 6.07e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 2.82e5iT - 8.32e11T^{2} \) |
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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
−11.11931696134309319323070138771, −10.14610778963907033164155709667, −9.737449388516902876984339027348, −9.153199943314064203685218319781, −7.67688113809492912019374906202, −6.27628927569048333044352937961, −5.52539535502899323519542281396, −4.36462612460853791184663432655, −3.32426842563673912247390678876, −2.20935507112322165355396735383,
0.00390699275237358958457974104, 0.932636827945121013616168458251, 2.52481284045350325382828233683, 3.02954303404379845264919463108, 5.39333996823951813921141870633, 5.93382200832951270051380238241, 7.08845668258323780667342000926, 7.70899235797105116212957413763, 8.882685779512758895504063308120, 10.13142778588776018285438450205