L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s + 4·11-s − 12-s + 2·13-s + 15-s + 16-s − 2·17-s − 18-s − 4·19-s − 20-s − 4·22-s + 8·23-s + 24-s + 25-s − 2·26-s − 27-s + 2·29-s − 30-s − 8·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.371·29-s − 0.182·30-s − 1.43·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.17624394082624, −14.63248557339359, −14.12948543470641, −13.24591118664109, −12.82777095634538, −12.40206010915689, −11.62395094340519, −11.26692213660119, −10.98839746155703, −10.38822111910646, −9.661869918342192, −9.171930863289349, −8.634159568958242, −8.305652824773085, −7.307157251509960, −7.034811132167405, −6.434899051590034, −6.000735786719525, −5.151123139470064, −4.545248824164590, −3.839723163269884, −3.328564766759941, −2.362422280347576, −1.534332434323971, −0.9279133791819109, 0,
0.9279133791819109, 1.534332434323971, 2.362422280347576, 3.328564766759941, 3.839723163269884, 4.545248824164590, 5.151123139470064, 6.000735786719525, 6.434899051590034, 7.034811132167405, 7.307157251509960, 8.305652824773085, 8.634159568958242, 9.171930863289349, 9.661869918342192, 10.38822111910646, 10.98839746155703, 11.26692213660119, 11.62395094340519, 12.40206010915689, 12.82777095634538, 13.24591118664109, 14.12948543470641, 14.63248557339359, 15.17624394082624