| L(s) = 1 | − 2·2-s + 2·4-s + 5-s − 2·10-s + 2·11-s − 4·16-s + 6·17-s + 3·19-s + 2·20-s − 4·22-s − 3·23-s − 4·25-s − 3·29-s + 3·31-s + 8·32-s − 12·34-s + 6·37-s − 6·38-s + 10·41-s − 43-s + 4·44-s + 6·46-s + 11·47-s + 8·50-s + 9·53-s + 2·55-s + 6·58-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s − 0.632·10-s + 0.603·11-s − 16-s + 1.45·17-s + 0.688·19-s + 0.447·20-s − 0.852·22-s − 0.625·23-s − 4/5·25-s − 0.557·29-s + 0.538·31-s + 1.41·32-s − 2.05·34-s + 0.986·37-s − 0.973·38-s + 1.56·41-s − 0.152·43-s + 0.603·44-s + 0.884·46-s + 1.60·47-s + 1.13·50-s + 1.23·53-s + 0.269·55-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.753118860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.753118860\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.02502129159043, −13.76684298628526, −13.05874162917617, −12.44213145884543, −11.87539276105352, −11.53492079612466, −10.89211919873337, −10.37053398052814, −9.837341687331106, −9.589771694398582, −9.137746702546847, −8.536584508329797, −7.958452773925929, −7.462705766786812, −7.253163628568175, −6.259739312197869, −5.938496655521941, −5.364109325388498, −4.504271425203283, −3.938315901389936, −3.241182580022897, −2.368108392515078, −1.894303635163325, −1.002902899441864, −0.6993927424866404,
0.6993927424866404, 1.002902899441864, 1.894303635163325, 2.368108392515078, 3.241182580022897, 3.938315901389936, 4.504271425203283, 5.364109325388498, 5.938496655521941, 6.259739312197869, 7.253163628568175, 7.462705766786812, 7.958452773925929, 8.536584508329797, 9.137746702546847, 9.589771694398582, 9.837341687331106, 10.37053398052814, 10.89211919873337, 11.53492079612466, 11.87539276105352, 12.44213145884543, 13.05874162917617, 13.76684298628526, 14.02502129159043
