L(s) = 1 | − 0.315·3-s + 2.25·5-s + 3.20·7-s − 2.90·9-s + 0.218·11-s − 4.17·13-s − 0.710·15-s − 4.68·17-s − 3.43·19-s − 1.01·21-s + 3.99·23-s + 0.0635·25-s + 1.86·27-s − 0.712·29-s − 1.04·31-s − 0.0688·33-s + 7.21·35-s + 1.70·37-s + 1.31·39-s + 3.18·41-s + 6.35·43-s − 6.52·45-s + 3.87·47-s + 3.28·49-s + 1.48·51-s − 10.8·53-s + 0.490·55-s + ⋯ |
L(s) = 1 | − 0.182·3-s + 1.00·5-s + 1.21·7-s − 0.966·9-s + 0.0657·11-s − 1.15·13-s − 0.183·15-s − 1.13·17-s − 0.789·19-s − 0.220·21-s + 0.833·23-s + 0.0127·25-s + 0.358·27-s − 0.132·29-s − 0.188·31-s − 0.0119·33-s + 1.21·35-s + 0.281·37-s + 0.211·39-s + 0.497·41-s + 0.969·43-s − 0.972·45-s + 0.565·47-s + 0.468·49-s + 0.207·51-s − 1.49·53-s + 0.0661·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 + 0.315T + 3T^{2} \) |
| 5 | \( 1 - 2.25T + 5T^{2} \) |
| 7 | \( 1 - 3.20T + 7T^{2} \) |
| 11 | \( 1 - 0.218T + 11T^{2} \) |
| 13 | \( 1 + 4.17T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 19 | \( 1 + 3.43T + 19T^{2} \) |
| 23 | \( 1 - 3.99T + 23T^{2} \) |
| 29 | \( 1 + 0.712T + 29T^{2} \) |
| 31 | \( 1 + 1.04T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 41 | \( 1 - 3.18T + 41T^{2} \) |
| 43 | \( 1 - 6.35T + 43T^{2} \) |
| 47 | \( 1 - 3.87T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 - 2.63T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 6.09T + 67T^{2} \) |
| 71 | \( 1 + 9.40T + 71T^{2} \) |
| 73 | \( 1 + 2.78T + 73T^{2} \) |
| 79 | \( 1 + 5.16T + 79T^{2} \) |
| 83 | \( 1 + 1.83T + 83T^{2} \) |
| 89 | \( 1 + 8.08T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.47929626886847559631619591270, −6.74578357969971850280992101775, −5.98885938078956573333093326862, −5.40069414922810508461805632154, −4.78479852578347896468035227216, −4.14508453765839415100035774354, −2.71136164798044342617013109858, −2.31255365548548628294672534861, −1.41546261645210980502326801104, 0,
1.41546261645210980502326801104, 2.31255365548548628294672534861, 2.71136164798044342617013109858, 4.14508453765839415100035774354, 4.78479852578347896468035227216, 5.40069414922810508461805632154, 5.98885938078956573333093326862, 6.74578357969971850280992101775, 7.47929626886847559631619591270