| L(s) = 1 | − 2.73·5-s − 7-s − 3.46·11-s + 4·13-s − 6.73·17-s + 19-s + 7.46·23-s + 2.46·25-s − 0.732·29-s + 3.46·31-s + 2.73·35-s − 0.535·37-s − 4.53·41-s + 4.92·43-s + 11.6·47-s + 49-s + 2.19·53-s + 9.46·55-s − 8·59-s + 11.4·61-s − 10.9·65-s − 4·67-s − 4.19·71-s + 7.46·73-s + 3.46·77-s + 1.46·79-s − 10.1·83-s + ⋯ |
| L(s) = 1 | − 1.22·5-s − 0.377·7-s − 1.04·11-s + 1.10·13-s − 1.63·17-s + 0.229·19-s + 1.55·23-s + 0.492·25-s − 0.135·29-s + 0.622·31-s + 0.461·35-s − 0.0881·37-s − 0.708·41-s + 0.751·43-s + 1.70·47-s + 0.142·49-s + 0.301·53-s + 1.27·55-s − 1.04·59-s + 1.46·61-s − 1.35·65-s − 0.488·67-s − 0.497·71-s + 0.873·73-s + 0.394·77-s + 0.164·79-s − 1.11·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 2.73T + 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 6.73T + 17T^{2} \) |
| 23 | \( 1 - 7.46T + 23T^{2} \) |
| 29 | \( 1 + 0.732T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + 0.535T + 37T^{2} \) |
| 41 | \( 1 + 4.53T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 2.19T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 11.4T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 4.19T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 - 1.46T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−7.32596981093123567795109009990, −6.82340138894936012718370614380, −6.03969934927840819380248966825, −5.20940538295844131186310197427, −4.45959312612735554465634134030, −3.85537965691787464342786601359, −3.08169825140157038054223834298, −2.36450367944707923194759578220, −0.991765080308852350426570996862, 0,
0.991765080308852350426570996862, 2.36450367944707923194759578220, 3.08169825140157038054223834298, 3.85537965691787464342786601359, 4.45959312612735554465634134030, 5.20940538295844131186310197427, 6.03969934927840819380248966825, 6.82340138894936012718370614380, 7.32596981093123567795109009990
