| L(s) = 1 | + 5.95·3-s + 6.64·5-s − 7·7-s + 8.52·9-s + 5.42·11-s + 24.2·13-s + 39.5·15-s − 60.9·17-s − 152.·19-s − 41.7·21-s − 170.·23-s − 80.8·25-s − 110.·27-s + 2.41·29-s − 96.4·31-s + 32.3·33-s − 46.5·35-s + 194.·37-s + 144.·39-s − 41·41-s − 171.·43-s + 56.6·45-s + 545.·47-s + 49·49-s − 363.·51-s − 209.·53-s + 36.0·55-s + ⋯ |
| L(s) = 1 | + 1.14·3-s + 0.594·5-s − 0.377·7-s + 0.315·9-s + 0.148·11-s + 0.517·13-s + 0.681·15-s − 0.869·17-s − 1.84·19-s − 0.433·21-s − 1.54·23-s − 0.646·25-s − 0.785·27-s + 0.0154·29-s − 0.558·31-s + 0.170·33-s − 0.224·35-s + 0.865·37-s + 0.593·39-s − 0.156·41-s − 0.608·43-s + 0.187·45-s + 1.69·47-s + 0.142·49-s − 0.997·51-s − 0.542·53-s + 0.0883·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 41 | \( 1 + 41T \) |
| good | 3 | \( 1 - 5.95T + 27T^{2} \) |
| 5 | \( 1 - 6.64T + 125T^{2} \) |
| 11 | \( 1 - 5.42T + 1.33e3T^{2} \) |
| 13 | \( 1 - 24.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 60.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 152.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 170.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 2.41T + 2.43e4T^{2} \) |
| 31 | \( 1 + 96.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 194.T + 5.06e4T^{2} \) |
| 43 | \( 1 + 171.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 545.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 209.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 356.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 420.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 705.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 552.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 376.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 790.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 223.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 532.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 322.T + 9.12e5T^{2} \) |
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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
−8.922675557189902247486841501735, −8.408819574287108633639862783857, −7.53718359208838054487966913455, −6.36319495880711097588773890151, −5.89494246791276377925013952818, −4.35828860134256451041253580918, −3.66921532437363776531681968688, −2.44063332803377766052442444725, −1.86997683750647171770750522105, 0,
1.86997683750647171770750522105, 2.44063332803377766052442444725, 3.66921532437363776531681968688, 4.35828860134256451041253580918, 5.89494246791276377925013952818, 6.36319495880711097588773890151, 7.53718359208838054487966913455, 8.408819574287108633639862783857, 8.922675557189902247486841501735
