| L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s − 3·10-s − 5·11-s + 16-s − 6·19-s + 3·20-s + 5·22-s + 2·23-s + 4·25-s + 9·29-s + 3·31-s − 32-s + 6·37-s + 6·38-s − 3·40-s + 6·41-s − 4·43-s − 5·44-s − 2·46-s + 6·47-s − 4·50-s − 3·53-s − 15·55-s − 9·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 0.948·10-s − 1.50·11-s + 1/4·16-s − 1.37·19-s + 0.670·20-s + 1.06·22-s + 0.417·23-s + 4/5·25-s + 1.67·29-s + 0.538·31-s − 0.176·32-s + 0.986·37-s + 0.973·38-s − 0.474·40-s + 0.937·41-s − 0.609·43-s − 0.753·44-s − 0.294·46-s + 0.875·47-s − 0.565·50-s − 0.412·53-s − 2.02·55-s − 1.18·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.004646340\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.004646340\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−12.71568254033483, −12.50560056184150, −11.99091785734647, −11.15241651176617, −10.79844442839296, −10.46697868766136, −10.14470816269842, −9.469825091950365, −9.347845160000813, −8.561933504043180, −8.248531385421109, −7.805751460290936, −7.227407042893669, −6.543617057379326, −6.256491355058809, −5.829099748042867, −5.226333703780320, −4.710986357094244, −4.268507124640218, −3.218273257043549, −2.734952117163741, −2.344056658414302, −1.873176832916642, −1.088402781986164, −0.4553621941123940,
0.4553621941123940, 1.088402781986164, 1.873176832916642, 2.344056658414302, 2.734952117163741, 3.218273257043549, 4.268507124640218, 4.710986357094244, 5.226333703780320, 5.829099748042867, 6.256491355058809, 6.543617057379326, 7.227407042893669, 7.805751460290936, 8.248531385421109, 8.561933504043180, 9.347845160000813, 9.469825091950365, 10.14470816269842, 10.46697868766136, 10.79844442839296, 11.15241651176617, 11.99091785734647, 12.50560056184150, 12.71568254033483
