| L(s) = 1 | + 3-s − 2·4-s + 5-s − 7-s + 9-s + 6·11-s − 2·12-s + 15-s + 4·16-s + 17-s − 2·19-s − 2·20-s − 21-s + 3·23-s + 25-s + 27-s + 2·28-s − 5·31-s + 6·33-s − 35-s − 2·36-s − 5·37-s − 9·41-s + 8·43-s − 12·44-s + 45-s + 9·47-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.80·11-s − 0.577·12-s + 0.258·15-s + 16-s + 0.242·17-s − 0.458·19-s − 0.447·20-s − 0.218·21-s + 0.625·23-s + 1/5·25-s + 0.192·27-s + 0.377·28-s − 0.898·31-s + 1.04·33-s − 0.169·35-s − 1/3·36-s − 0.821·37-s − 1.40·41-s + 1.21·43-s − 1.80·44-s + 0.149·45-s + 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.389220213\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.389220213\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.68842742785498, −12.35061280516577, −11.96986826044716, −11.27441337146077, −10.77937680267197, −10.21061489150913, −9.825692208915838, −9.344265601468959, −9.006076973304883, −8.697295381448445, −8.329633373114098, −7.441637213112316, −7.181210604076452, −6.597816044369832, −6.067300320224261, −5.627226332465011, −5.013882894684662, −4.444290055784137, −4.003283218091453, −3.482418466606237, −3.203027212231884, −2.277165523123533, −1.724835449305796, −1.110763586968949, −0.5291435655207801,
0.5291435655207801, 1.110763586968949, 1.724835449305796, 2.277165523123533, 3.203027212231884, 3.482418466606237, 4.003283218091453, 4.444290055784137, 5.013882894684662, 5.627226332465011, 6.067300320224261, 6.597816044369832, 7.181210604076452, 7.441637213112316, 8.329633373114098, 8.697295381448445, 9.006076973304883, 9.344265601468959, 9.825692208915838, 10.21061489150913, 10.77937680267197, 11.27441337146077, 11.96986826044716, 12.35061280516577, 12.68842742785498
