| L(s) = 1 | + 1.97·2-s − 2.64·3-s + 1.88·4-s − 5-s − 5.22·6-s + 0.783·7-s − 0.221·8-s + 4.01·9-s − 1.97·10-s − 0.568·11-s − 5.00·12-s + 0.00611·13-s + 1.54·14-s + 2.64·15-s − 4.21·16-s − 2.96·17-s + 7.92·18-s − 1.03·19-s − 1.88·20-s − 2.07·21-s − 1.12·22-s + 8.65·23-s + 0.585·24-s + 25-s + 0.0120·26-s − 2.69·27-s + 1.47·28-s + ⋯ |
| L(s) = 1 | + 1.39·2-s − 1.52·3-s + 0.943·4-s − 0.447·5-s − 2.13·6-s + 0.296·7-s − 0.0781·8-s + 1.33·9-s − 0.623·10-s − 0.171·11-s − 1.44·12-s + 0.00169·13-s + 0.412·14-s + 0.684·15-s − 1.05·16-s − 0.719·17-s + 1.86·18-s − 0.237·19-s − 0.422·20-s − 0.453·21-s − 0.239·22-s + 1.80·23-s + 0.119·24-s + 0.200·25-s + 0.00236·26-s − 0.519·27-s + 0.279·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 2 | \( 1 - 1.97T + 2T^{2} \) |
| 3 | \( 1 + 2.64T + 3T^{2} \) |
| 7 | \( 1 - 0.783T + 7T^{2} \) |
| 11 | \( 1 + 0.568T + 11T^{2} \) |
| 13 | \( 1 - 0.00611T + 13T^{2} \) |
| 17 | \( 1 + 2.96T + 17T^{2} \) |
| 19 | \( 1 + 1.03T + 19T^{2} \) |
| 23 | \( 1 - 8.65T + 23T^{2} \) |
| 29 | \( 1 - 4.06T + 29T^{2} \) |
| 31 | \( 1 - 9.38T + 31T^{2} \) |
| 41 | \( 1 + 4.13T + 41T^{2} \) |
| 43 | \( 1 + 7.06T + 43T^{2} \) |
| 47 | \( 1 - 5.52T + 47T^{2} \) |
| 53 | \( 1 + 9.14T + 53T^{2} \) |
| 59 | \( 1 + 9.24T + 59T^{2} \) |
| 61 | \( 1 - 9.98T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 3.92T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 0.861T + 83T^{2} \) |
| 89 | \( 1 - 0.287T + 89T^{2} \) |
| 97 | \( 1 - 4.33T + 97T^{2} \) |
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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
−7.12718673593462796122428696935, −6.56071797422399626917942214084, −6.18790934889163734009142572959, −5.19674992591166665038328778981, −4.83600754210555543441014682265, −4.42593776852904379553017970309, −3.38704863606186054686165003785, −2.58779794398200016758136861443, −1.18799022329610421454209192668, 0,
1.18799022329610421454209192668, 2.58779794398200016758136861443, 3.38704863606186054686165003785, 4.42593776852904379553017970309, 4.83600754210555543441014682265, 5.19674992591166665038328778981, 6.18790934889163734009142572959, 6.56071797422399626917942214084, 7.12718673593462796122428696935
