L(s) = 1 | + 2-s + 4-s − 2·7-s + 8-s + 11-s + 6·13-s − 2·14-s + 16-s + 2·17-s + 6·19-s + 22-s − 4·23-s + 6·26-s − 2·28-s + 8·29-s − 9·31-s + 32-s + 2·34-s + 2·37-s + 6·38-s − 4·41-s − 43-s + 44-s − 4·46-s − 3·49-s + 6·52-s − 12·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s + 0.301·11-s + 1.66·13-s − 0.534·14-s + 1/4·16-s + 0.485·17-s + 1.37·19-s + 0.213·22-s − 0.834·23-s + 1.17·26-s − 0.377·28-s + 1.48·29-s − 1.61·31-s + 0.176·32-s + 0.342·34-s + 0.328·37-s + 0.973·38-s − 0.624·41-s − 0.152·43-s + 0.150·44-s − 0.589·46-s − 3/7·49-s + 0.832·52-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 127 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−14.41586252291232, −13.97875128396015, −13.73352296291256, −13.16822870272750, −12.47583063363838, −12.35116543052468, −11.51593842346518, −11.23065687996169, −10.63165369023062, −9.979439719518610, −9.593532360047009, −8.946237607710960, −8.357475205634223, −7.782418510259818, −7.186160357545459, −6.576385704083297, −5.993985830786384, −5.810434324196815, −4.955923793162989, −4.363698447622475, −3.536846903521842, −3.370441262110232, −2.733009798760124, −1.571340929916381, −1.243227643460770, 0,
1.243227643460770, 1.571340929916381, 2.733009798760124, 3.370441262110232, 3.536846903521842, 4.363698447622475, 4.955923793162989, 5.810434324196815, 5.993985830786384, 6.576385704083297, 7.186160357545459, 7.782418510259818, 8.357475205634223, 8.946237607710960, 9.593532360047009, 9.979439719518610, 10.63165369023062, 11.23065687996169, 11.51593842346518, 12.35116543052468, 12.47583063363838, 13.16822870272750, 13.73352296291256, 13.97875128396015, 14.41586252291232