| L(s) = 1 | − 2-s + 4-s − 1.80·5-s − 8-s + 9-s + 1.80·10-s − 0.445·13-s + 16-s − 1.24·17-s − 18-s − 1.80·20-s + 2.24·25-s + 0.445·26-s − 32-s + 1.24·34-s + 36-s + 0.445·37-s + 1.80·40-s + 0.445·41-s − 1.80·45-s + 49-s − 2.24·50-s − 0.445·52-s + 1.24·53-s − 1.24·61-s + 64-s + 0.801·65-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 1.80·5-s − 8-s + 9-s + 1.80·10-s − 0.445·13-s + 16-s − 1.24·17-s − 18-s − 1.80·20-s + 2.24·25-s + 0.445·26-s − 32-s + 1.24·34-s + 36-s + 0.445·37-s + 1.80·40-s + 0.445·41-s − 1.80·45-s + 49-s − 2.24·50-s − 0.445·52-s + 1.24·53-s − 1.24·61-s + 64-s + 0.801·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5157527059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5157527059\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - T^{2} \) |
| 5 | \( 1 + 1.80T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 0.445T + T^{2} \) |
| 17 | \( 1 + 1.24T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 0.445T + T^{2} \) |
| 41 | \( 1 - 0.445T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - 1.24T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.24T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.24T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.80T + T^{2} \) |
| 97 | \( 1 - 0.445T + T^{2} \) |
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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
−8.745433892086909148771692172736, −8.030150694646863650413342224880, −7.28972062358337463590929590647, −7.10100573146195452117070645446, −6.09753885077374383339514090944, −4.72500517201083121061096553805, −4.12748198461772658322868111250, −3.21068963080272709495590806761, −2.11863338502720460257551192956, −0.71777170660935102793492447871,
0.71777170660935102793492447871, 2.11863338502720460257551192956, 3.21068963080272709495590806761, 4.12748198461772658322868111250, 4.72500517201083121061096553805, 6.09753885077374383339514090944, 7.10100573146195452117070645446, 7.28972062358337463590929590647, 8.030150694646863650413342224880, 8.745433892086909148771692172736
