| L(s) = 1 | − 1.95·2-s + 1.82·4-s + 5-s − 3.56·7-s + 0.340·8-s − 1.95·10-s − 5.56·11-s + 5.26·13-s + 6.96·14-s − 4.31·16-s − 1.40·17-s + 19-s + 1.82·20-s + 10.8·22-s + 6.96·23-s + 25-s − 10.3·26-s − 6.50·28-s − 1.40·29-s + 1.75·31-s + 7.76·32-s + 2.75·34-s − 3.56·35-s + 3.61·37-s − 1.95·38-s + 0.340·40-s − 4.34·41-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 0.912·4-s + 0.447·5-s − 1.34·7-s + 0.120·8-s − 0.618·10-s − 1.67·11-s + 1.46·13-s + 1.86·14-s − 1.07·16-s − 0.341·17-s + 0.229·19-s + 0.408·20-s + 2.31·22-s + 1.45·23-s + 0.200·25-s − 2.02·26-s − 1.22·28-s − 0.261·29-s + 0.315·31-s + 1.37·32-s + 0.471·34-s − 0.602·35-s + 0.594·37-s − 0.317·38-s + 0.0539·40-s − 0.679·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6031018279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6031018279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.95T + 2T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 + 5.56T + 11T^{2} \) |
| 13 | \( 1 - 5.26T + 13T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 23 | \( 1 - 6.96T + 23T^{2} \) |
| 29 | \( 1 + 1.40T + 29T^{2} \) |
| 31 | \( 1 - 1.75T + 31T^{2} \) |
| 37 | \( 1 - 3.61T + 37T^{2} \) |
| 41 | \( 1 + 4.34T + 41T^{2} \) |
| 43 | \( 1 + 3.56T + 43T^{2} \) |
| 47 | \( 1 - 8.26T + 47T^{2} \) |
| 53 | \( 1 - 7.61T + 53T^{2} \) |
| 59 | \( 1 + 9.47T + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 + 4.76T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 6.59T + 73T^{2} \) |
| 79 | \( 1 - 5.47T + 79T^{2} \) |
| 83 | \( 1 - 4.15T + 83T^{2} \) |
| 89 | \( 1 - 9.23T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−10.15033171056742891138433483499, −9.267561546844698726660910101551, −8.716238895055914956669405812424, −7.81450863242715631822709248237, −6.90987868690790929687096743717, −6.11118087356776791733753571403, −5.01897693061581487824495416711, −3.43143893903231896065012408166, −2.36458969233167643172870884214, −0.75670321910440066947189776463,
0.75670321910440066947189776463, 2.36458969233167643172870884214, 3.43143893903231896065012408166, 5.01897693061581487824495416711, 6.11118087356776791733753571403, 6.90987868690790929687096743717, 7.81450863242715631822709248237, 8.716238895055914956669405812424, 9.267561546844698726660910101551, 10.15033171056742891138433483499
