| L(s) = 1 | + 5-s − 3·7-s + 2·13-s + 2·17-s − 19-s + 2·23-s + 25-s − 8·29-s + 5·31-s − 3·35-s − 11·37-s + 6·41-s + 8·43-s − 8·47-s + 2·49-s − 2·53-s − 4·59-s + 3·61-s + 2·65-s − 9·67-s + 6·71-s + 11·73-s + 9·79-s + 8·83-s + 2·85-s − 6·89-s − 6·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s + 0.554·13-s + 0.485·17-s − 0.229·19-s + 0.417·23-s + 1/5·25-s − 1.48·29-s + 0.898·31-s − 0.507·35-s − 1.80·37-s + 0.937·41-s + 1.21·43-s − 1.16·47-s + 2/7·49-s − 0.274·53-s − 0.520·59-s + 0.384·61-s + 0.248·65-s − 1.09·67-s + 0.712·71-s + 1.28·73-s + 1.01·79-s + 0.878·83-s + 0.216·85-s − 0.635·89-s − 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−15.94649314634645, −15.24125222512358, −14.85300546471676, −13.96556967055755, −13.75571913347793, −13.04499788916545, −12.62740561224472, −12.20276501489825, −11.35076521138479, −10.83152897615475, −10.30605891935418, −9.625072192520044, −9.297862249217206, −8.699718752813659, −7.943465698998573, −7.333199831726145, −6.570558669257963, −6.254407166322172, −5.540556969722149, −4.975696010210936, −3.982805970501062, −3.464846193951560, −2.810375204295224, −1.967194898074376, −1.069123508528895, 0,
1.069123508528895, 1.967194898074376, 2.810375204295224, 3.464846193951560, 3.982805970501062, 4.975696010210936, 5.540556969722149, 6.254407166322172, 6.570558669257963, 7.333199831726145, 7.943465698998573, 8.699718752813659, 9.297862249217206, 9.625072192520044, 10.30605891935418, 10.83152897615475, 11.35076521138479, 12.20276501489825, 12.62740561224472, 13.04499788916545, 13.75571913347793, 13.96556967055755, 14.85300546471676, 15.24125222512358, 15.94649314634645
