| L(s) = 1 | − 1.80·2-s + 2.24·4-s + 7-s − 2.24·8-s + 9-s + 1.24·11-s − 1.80·14-s + 1.80·16-s − 1.80·18-s − 2.24·22-s − 0.445·23-s + 25-s + 2.24·28-s − 1.80·29-s − 1.00·32-s + 2.24·36-s − 0.445·37-s − 1.80·43-s + 2.80·44-s + 0.801·46-s + 49-s − 1.80·50-s − 0.445·53-s − 2.24·56-s + 3.24·58-s + 63-s − 0.445·67-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 2.24·4-s + 7-s − 2.24·8-s + 9-s + 1.24·11-s − 1.80·14-s + 1.80·16-s − 1.80·18-s − 2.24·22-s − 0.445·23-s + 25-s + 2.24·28-s − 1.80·29-s − 1.00·32-s + 2.24·36-s − 0.445·37-s − 1.80·43-s + 2.80·44-s + 0.801·46-s + 49-s − 1.80·50-s − 0.445·53-s − 2.24·56-s + 3.24·58-s + 63-s − 0.445·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6032097230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6032097230\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.80T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 - 1.24T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 0.445T + T^{2} \) |
| 29 | \( 1 + 1.80T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + 0.445T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + 1.80T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 0.445T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 0.445T + T^{2} \) |
| 71 | \( 1 + 0.445T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 1.24T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.828384687608852769845120182048, −9.108537370802856126684910944867, −8.499191145247006009455017776763, −7.60513379721912829456053608686, −7.05251752677127003818424448919, −6.22174566889859470914756399726, −4.85313061377590386662183590435, −3.65536776975584169137223988453, −1.97640951424038817062552751330, −1.30827570104108469382417681351,
1.30827570104108469382417681351, 1.97640951424038817062552751330, 3.65536776975584169137223988453, 4.85313061377590386662183590435, 6.22174566889859470914756399726, 7.05251752677127003818424448919, 7.60513379721912829456053608686, 8.499191145247006009455017776763, 9.108537370802856126684910944867, 9.828384687608852769845120182048
