| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 3·9-s − 10-s − 4·11-s + 6·13-s + 16-s − 3·18-s − 20-s − 4·22-s + 25-s + 6·26-s − 6·29-s + 8·31-s + 32-s − 3·36-s + 10·37-s − 40-s + 2·41-s + 4·43-s − 4·44-s + 3·45-s − 8·47-s + 50-s + 6·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 9-s − 0.316·10-s − 1.20·11-s + 1.66·13-s + 1/4·16-s − 0.707·18-s − 0.223·20-s − 0.852·22-s + 1/5·25-s + 1.17·26-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 1/2·36-s + 1.64·37-s − 0.158·40-s + 0.312·41-s + 0.609·43-s − 0.603·44-s + 0.447·45-s − 1.16·47-s + 0.141·50-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.56906476922445, −13.13801783813382, −12.92068874189312, −12.13125657331026, −11.78325289943517, −11.14198055556632, −10.95513158319005, −10.61166668391830, −9.769291181990474, −9.321528337078299, −8.614179612361521, −8.110038904916629, −7.941466626757171, −7.313959185762944, −6.467735063026448, −6.188831678528398, −5.674502303695478, −5.189211907106571, −4.582652371517277, −3.978342293957579, −3.507987305312335, −2.773622867015106, −2.610576870708447, −1.603629055576995, −0.8670152567736104, 0,
0.8670152567736104, 1.603629055576995, 2.610576870708447, 2.773622867015106, 3.507987305312335, 3.978342293957579, 4.582652371517277, 5.189211907106571, 5.674502303695478, 6.188831678528398, 6.467735063026448, 7.313959185762944, 7.941466626757171, 8.110038904916629, 8.614179612361521, 9.321528337078299, 9.769291181990474, 10.61166668391830, 10.95513158319005, 11.14198055556632, 11.78325289943517, 12.13125657331026, 12.92068874189312, 13.13801783813382, 13.56906476922445
