| L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 3·9-s + 11-s − 14-s + 16-s − 2·17-s − 3·18-s + 3·19-s + 22-s + 3·23-s − 5·25-s − 28-s − 3·29-s + 4·31-s + 32-s − 2·34-s − 3·36-s − 2·37-s + 3·38-s + 6·41-s + 43-s + 44-s + 3·46-s + 3·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 9-s + 0.301·11-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 0.688·19-s + 0.213·22-s + 0.625·23-s − 25-s − 0.188·28-s − 0.557·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s − 0.328·37-s + 0.486·38-s + 0.937·41-s + 0.152·43-s + 0.150·44-s + 0.442·46-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−15.47164140540995, −15.02681847638622, −14.41852914970733, −14.00714229784389, −13.42577433730644, −13.07902976876355, −12.32954513224216, −11.77691999006290, −11.47499035952529, −10.84961118961589, −10.24593208253038, −9.560005477250310, −8.978742267885733, −8.521838702282211, −7.541939600361533, −7.366675254843577, −6.311067698130772, −6.096586664062398, −5.412635916139514, −4.771573372147429, −4.057152065453532, −3.388099632975368, −2.810757887952815, −2.134621400232531, −1.122766604981150, 0,
1.122766604981150, 2.134621400232531, 2.810757887952815, 3.388099632975368, 4.057152065453532, 4.771573372147429, 5.412635916139514, 6.096586664062398, 6.311067698130772, 7.366675254843577, 7.541939600361533, 8.521838702282211, 8.978742267885733, 9.560005477250310, 10.24593208253038, 10.84961118961589, 11.47499035952529, 11.77691999006290, 12.32954513224216, 13.07902976876355, 13.42577433730644, 14.00714229784389, 14.41852914970733, 15.02681847638622, 15.47164140540995
