| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 3·9-s − 10-s − 16-s + 6·17-s + 3·18-s + 2·19-s − 20-s − 5·23-s + 25-s − 29-s + 8·31-s − 5·32-s − 6·34-s + 3·36-s + 6·37-s − 2·38-s + 3·40-s − 7·41-s − 5·43-s − 3·45-s + 5·46-s − 47-s − 50-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 − 1/4·16-s + 1.45·17-s + 0.707·18-s + 0.458·19-s − 0.223·20-s − 1.04·23-s + 1/5·25-s − 0.185·29-s + 1.43·31-s − 0.883·32-s − 1.02·34-s + 1/2·36-s + 0.986·37-s − 0.324·38-s + 0.474·40-s − 1.09·41-s − 0.762·43-s − 0.447·45-s + 0.737·46-s − 0.145·47-s − 0.141·50-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 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.45260228513522, −14.76313083154430, −14.20155818103287, −13.92272099547313, −13.49202194906345, −12.71349760268113, −12.24903610971385, −11.50364549662109, −11.22307111515392, −10.16597517684017, −10.02360871739569, −9.633328436256611, −8.792535973846386, −8.417075555636302, −7.893850400358327, −7.431089377373155, −6.480201144862977, −5.954301844625429, −5.308537935019538, −4.852962183960588, −3.978260707950401, −3.285435258744658, −2.604481085069553, −1.643537126117008, −0.9466780566658955, 0,
0.9466780566658955, 1.643537126117008, 2.604481085069553, 3.285435258744658, 3.978260707950401, 4.852962183960588, 5.308537935019538, 5.954301844625429, 6.480201144862977, 7.431089377373155, 7.893850400358327, 8.417075555636302, 8.792535973846386, 9.633328436256611, 10.02360871739569, 10.16597517684017, 11.22307111515392, 11.50364549662109, 12.24903610971385, 12.71349760268113, 13.49202194906345, 13.92272099547313, 14.20155818103287, 14.76313083154430, 15.45260228513522
