| L(s) = 1 | − 2-s + 4-s + 3·5-s − 7-s − 8-s − 3·10-s − 6·11-s + 2·13-s + 14-s + 16-s + 6·17-s + 19-s + 3·20-s + 6·22-s + 6·23-s + 4·25-s − 2·26-s − 28-s − 10·31-s − 32-s − 6·34-s − 3·35-s − 4·37-s − 38-s − 3·40-s − 12·41-s − 43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.377·7-s − 0.353·8-s − 0.948·10-s − 1.80·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.229·19-s + 0.670·20-s + 1.27·22-s + 1.25·23-s + 4/5·25-s − 0.392·26-s − 0.188·28-s − 1.79·31-s − 0.176·32-s − 1.02·34-s − 0.507·35-s − 0.657·37-s − 0.162·38-s − 0.474·40-s − 1.87·41-s − 0.152·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| 113 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.03643299392571, −14.70774092219809, −13.85305330487780, −13.59275262797421, −12.87530387241293, −12.72576414629821, −11.98235349426866, −11.09137403220969, −10.77051292588115, −10.23787031181694, −9.805876868688849, −9.403676210191978, −8.739713513177763, −8.187980588075257, −7.592289386105282, −7.049169245621756, −6.420250331927936, −5.731549629319960, −5.276015267177924, −5.012763387092567, −3.464144130127523, −3.247261887827199, −2.377360369538348, −1.797631290886820, −1.035957068347279, 0,
1.035957068347279, 1.797631290886820, 2.377360369538348, 3.247261887827199, 3.464144130127523, 5.012763387092567, 5.276015267177924, 5.731549629319960, 6.420250331927936, 7.049169245621756, 7.592289386105282, 8.187980588075257, 8.739713513177763, 9.403676210191978, 9.805876868688849, 10.23787031181694, 10.77051292588115, 11.09137403220969, 11.98235349426866, 12.72576414629821, 12.87530387241293, 13.59275262797421, 13.85305330487780, 14.70774092219809, 15.03643299392571
