| L(s) = 1 | − 2-s − 4-s − 2·5-s + 3·8-s + 2·10-s − 6·13-s − 16-s − 6·17-s + 19-s + 2·20-s − 4·23-s − 25-s + 6·26-s − 2·29-s − 8·31-s − 5·32-s + 6·34-s − 10·37-s − 38-s − 6·40-s − 2·41-s − 4·43-s + 4·46-s + 12·47-s + 50-s + 6·52-s + 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.06·8-s + 0.632·10-s − 1.66·13-s − 1/4·16-s − 1.45·17-s + 0.229·19-s + 0.447·20-s − 0.834·23-s − 1/5·25-s + 1.17·26-s − 0.371·29-s − 1.43·31-s − 0.883·32-s + 1.02·34-s − 1.64·37-s − 0.162·38-s − 0.948·40-s − 0.312·41-s − 0.609·43-s + 0.589·46-s + 1.75·47-s + 0.141·50-s + 0.832·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.34752150402139467995584348966, −6.78418502492955034999702547464, −5.56436473344957075485364042125, −4.94046317313085415521526307990, −4.20789833671809510725213788268, −3.70760760284754853820305649861, −2.47817878081925227300773446057, −1.65980873553023464631625543363, 0, 0,
1.65980873553023464631625543363, 2.47817878081925227300773446057, 3.70760760284754853820305649861, 4.20789833671809510725213788268, 4.94046317313085415521526307990, 5.56436473344957075485364042125, 6.78418502492955034999702547464, 7.34752150402139467995584348966
