| L(s) = 1 | + 1.51·3-s + 5-s + 1.32·7-s − 0.707·9-s − 3.32·11-s − 1.80·13-s + 1.51·15-s + 2·17-s + 19-s + 2·21-s − 5.32·23-s + 25-s − 5.61·27-s − 5.02·29-s − 5.02·31-s − 5.02·33-s + 1.32·35-s − 0.193·37-s − 2.73·39-s + 0.971·41-s + 4.34·43-s − 0.707·45-s + 1.32·47-s − 5.25·49-s + 3.02·51-s − 3.80·53-s − 3.32·55-s + ⋯ |
| L(s) = 1 | + 0.874·3-s + 0.447·5-s + 0.499·7-s − 0.235·9-s − 1.00·11-s − 0.501·13-s + 0.390·15-s + 0.485·17-s + 0.229·19-s + 0.436·21-s − 1.10·23-s + 0.200·25-s − 1.08·27-s − 0.933·29-s − 0.903·31-s − 0.875·33-s + 0.223·35-s − 0.0317·37-s − 0.438·39-s + 0.151·41-s + 0.663·43-s − 0.105·45-s + 0.192·47-s − 0.750·49-s + 0.424·51-s − 0.522·53-s − 0.447·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.51T + 3T^{2} \) |
| 7 | \( 1 - 1.32T + 7T^{2} \) |
| 11 | \( 1 + 3.32T + 11T^{2} \) |
| 13 | \( 1 + 1.80T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 23 | \( 1 + 5.32T + 23T^{2} \) |
| 29 | \( 1 + 5.02T + 29T^{2} \) |
| 31 | \( 1 + 5.02T + 31T^{2} \) |
| 37 | \( 1 + 0.193T + 37T^{2} \) |
| 41 | \( 1 - 0.971T + 41T^{2} \) |
| 43 | \( 1 - 4.34T + 43T^{2} \) |
| 47 | \( 1 - 1.32T + 47T^{2} \) |
| 53 | \( 1 + 3.80T + 53T^{2} \) |
| 59 | \( 1 - 8.05T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 - 5.02T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 + 2.29T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 1.16T + 97T^{2} \) |
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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
−7.67321656093834410122725919125, −7.43104157976422137842862458159, −6.14278695914384389089726935806, −5.56450529161761230498693839451, −4.90139360595583994505550941392, −3.91260233418145068974188021734, −3.07708777418381273014530681750, −2.35221382086916147270827997690, −1.64744287366201213112185112984, 0,
1.64744287366201213112185112984, 2.35221382086916147270827997690, 3.07708777418381273014530681750, 3.91260233418145068974188021734, 4.90139360595583994505550941392, 5.56450529161761230498693839451, 6.14278695914384389089726935806, 7.43104157976422137842862458159, 7.67321656093834410122725919125
