| L(s) = 1 | + 2-s − 4-s − 5-s − 3·8-s − 3·9-s − 10-s − 6·13-s − 16-s + 6·17-s − 3·18-s − 4·19-s + 20-s − 8·23-s + 25-s − 6·26-s + 10·29-s + 4·31-s + 5·32-s + 6·34-s + 3·36-s + 6·37-s − 4·38-s + 3·40-s − 10·41-s − 4·43-s + 3·45-s − 8·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.06·8-s − 9-s − 0.316·10-s − 1.66·13-s − 1/4·16-s + 1.45·17-s − 0.707·18-s − 0.917·19-s + 0.223·20-s − 1.66·23-s + 1/5·25-s − 1.17·26-s + 1.85·29-s + 0.718·31-s + 0.883·32-s + 1.02·34-s + 1/2·36-s + 0.986·37-s − 0.648·38-s + 0.474·40-s − 1.56·41-s − 0.609·43-s + 0.447·45-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.16679635115742, −14.78653416904217, −14.33274430915070, −13.88583794602475, −13.52156424934345, −12.49323767680900, −12.25358362239825, −11.98769193318958, −11.44616630451718, −10.44350152859630, −9.998623646906526, −9.646133534228225, −8.738961598926740, −8.242325671950773, −7.950345171396573, −7.121388920573685, −6.262531885910062, −5.964521350824423, −5.020235200875935, −4.870383582021150, −4.065475060076043, −3.423391816250314, −2.776624652695021, −2.208886948047193, −0.7770892604278816, 0,
0.7770892604278816, 2.208886948047193, 2.776624652695021, 3.423391816250314, 4.065475060076043, 4.870383582021150, 5.020235200875935, 5.964521350824423, 6.262531885910062, 7.121388920573685, 7.950345171396573, 8.242325671950773, 8.738961598926740, 9.646133534228225, 9.998623646906526, 10.44350152859630, 11.44616630451718, 11.98769193318958, 12.25358362239825, 12.49323767680900, 13.52156424934345, 13.88583794602475, 14.33274430915070, 14.78653416904217, 15.16679635115742
