| L(s) = 1 | − 1.55·2-s + 4.96·3-s − 5.58·4-s − 7.71·6-s − 7·7-s + 21.1·8-s − 2.38·9-s + 29.8·11-s − 27.6·12-s − 90.7·13-s + 10.8·14-s + 11.7·16-s − 29.5·17-s + 3.70·18-s − 62.3·19-s − 34.7·21-s − 46.3·22-s − 90.6·23-s + 104.·24-s + 141.·26-s − 145.·27-s + 39.0·28-s + 193.·29-s − 152.·31-s − 187.·32-s + 147.·33-s + 45.9·34-s + ⋯ |
| L(s) = 1 | − 0.549·2-s + 0.954·3-s − 0.697·4-s − 0.525·6-s − 0.377·7-s + 0.933·8-s − 0.0882·9-s + 0.817·11-s − 0.666·12-s − 1.93·13-s + 0.207·14-s + 0.184·16-s − 0.421·17-s + 0.0485·18-s − 0.752·19-s − 0.360·21-s − 0.449·22-s − 0.821·23-s + 0.891·24-s + 1.06·26-s − 1.03·27-s + 0.263·28-s + 1.23·29-s − 0.881·31-s − 1.03·32-s + 0.780·33-s + 0.232·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 1.55T + 8T^{2} \) |
| 3 | \( 1 - 4.96T + 27T^{2} \) |
| 11 | \( 1 - 29.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 90.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 29.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 62.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 90.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 193.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 152.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 102.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 266.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 387.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 152.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 81.5T + 1.48e5T^{2} \) |
| 59 | \( 1 + 235.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 510.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 347.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 317.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 709.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 503.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 482.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 481.T + 9.12e5T^{2} \) |
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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
−11.87090338807152925565847919254, −10.29523773581558706612515267513, −9.556645178497382588969375651863, −8.783426614830285971857754400643, −7.936645098058105761761755501884, −6.76912014646939290192206236008, −4.99134747386727011341159491398, −3.73887235298642340680868516013, −2.18617878655924310818044910082, 0,
2.18617878655924310818044910082, 3.73887235298642340680868516013, 4.99134747386727011341159491398, 6.76912014646939290192206236008, 7.936645098058105761761755501884, 8.783426614830285971857754400643, 9.556645178497382588969375651863, 10.29523773581558706612515267513, 11.87090338807152925565847919254
