| L(s) = 1 | + 23.5·3-s + 27.0·5-s + 36.8·7-s + 312.·9-s − 492.·11-s − 13.9·13-s + 636.·15-s + 1.07e3·17-s − 1.50e3·19-s + 867.·21-s − 3.47e3·23-s − 2.39e3·25-s + 1.63e3·27-s − 2.94e3·29-s − 1.97e3·31-s − 1.16e4·33-s + 994.·35-s + 867.·37-s − 328.·39-s + 6.69e3·41-s − 8.05e3·43-s + 8.44e3·45-s − 203.·47-s − 1.54e4·49-s + 2.53e4·51-s − 1.15e4·53-s − 1.33e4·55-s + ⋯ |
| L(s) = 1 | + 1.51·3-s + 0.483·5-s + 0.283·7-s + 1.28·9-s − 1.22·11-s − 0.0228·13-s + 0.730·15-s + 0.900·17-s − 0.958·19-s + 0.429·21-s − 1.37·23-s − 0.766·25-s + 0.432·27-s − 0.650·29-s − 0.369·31-s − 1.85·33-s + 0.137·35-s + 0.104·37-s − 0.0345·39-s + 0.621·41-s − 0.664·43-s + 0.621·45-s − 0.0134·47-s − 0.919·49-s + 1.36·51-s − 0.566·53-s − 0.593·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 - 23.5T + 243T^{2} \) |
| 5 | \( 1 - 27.0T + 3.12e3T^{2} \) |
| 7 | \( 1 - 36.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + 492.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 13.9T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.07e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.50e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.47e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.97e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 867.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.69e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.05e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 203.T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.15e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.44e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.39e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.04e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.45e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.03e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.14e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.87e3T + 8.58e9T^{2} \) |
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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
−8.694438799439400061545586260410, −7.914714182696203336604456777829, −7.61998196911429164081146304330, −6.22676939297970885023650117936, −5.32683233911866726225968197267, −4.17471177542634334635109289042, −3.26495187003277379052577508632, −2.30945564133706563024575811370, −1.71294924632650491245736383942, 0,
1.71294924632650491245736383942, 2.30945564133706563024575811370, 3.26495187003277379052577508632, 4.17471177542634334635109289042, 5.32683233911866726225968197267, 6.22676939297970885023650117936, 7.61998196911429164081146304330, 7.914714182696203336604456777829, 8.694438799439400061545586260410
