| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 6·11-s + 16-s + 17-s + 20-s − 6·22-s + 6·23-s + 25-s − 4·31-s + 32-s + 34-s + 10·37-s + 40-s − 8·41-s − 4·43-s − 6·44-s + 6·46-s − 7·49-s + 50-s − 10·53-s − 6·55-s + 4·59-s − 8·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 1.80·11-s + 1/4·16-s + 0.242·17-s + 0.223·20-s − 1.27·22-s + 1.25·23-s + 1/5·25-s − 0.718·31-s + 0.176·32-s + 0.171·34-s + 1.64·37-s + 0.158·40-s − 1.24·41-s − 0.609·43-s − 0.904·44-s + 0.884·46-s − 49-s + 0.141·50-s − 1.37·53-s − 0.809·55-s + 0.520·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−12.99260768126240, −12.81715258309027, −12.29294205387846, −11.70082435196275, −11.06185014808891, −10.89272870063721, −10.42411842658667, −9.815914248642127, −9.489494537497639, −8.907718429319972, −8.090953320338962, −7.975890932148184, −7.423451109566532, −6.775769375772268, −6.403217263967658, −5.809709180342449, −5.277113472995863, −4.910310219124973, −4.664605174227979, −3.654056237838744, −3.281150173866616, −2.721986889173314, −2.254757685823336, −1.636957747263956, −0.8487702763057358, 0,
0.8487702763057358, 1.636957747263956, 2.254757685823336, 2.721986889173314, 3.281150173866616, 3.654056237838744, 4.664605174227979, 4.910310219124973, 5.277113472995863, 5.809709180342449, 6.403217263967658, 6.775769375772268, 7.423451109566532, 7.975890932148184, 8.090953320338962, 8.907718429319972, 9.489494537497639, 9.815914248642127, 10.42411842658667, 10.89272870063721, 11.06185014808891, 11.70082435196275, 12.29294205387846, 12.81715258309027, 12.99260768126240
