| L(s) = 1 | + 5-s − 4·7-s − 2·17-s + 25-s + 2·29-s + 4·31-s − 4·35-s − 6·37-s − 6·41-s + 4·43-s − 4·47-s + 9·49-s + 10·53-s − 2·61-s − 8·67-s + 4·71-s + 6·73-s − 8·79-s + 8·83-s − 2·85-s − 6·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s − 0.485·17-s + 1/5·25-s + 0.371·29-s + 0.718·31-s − 0.676·35-s − 0.986·37-s − 0.937·41-s + 0.609·43-s − 0.583·47-s + 9/7·49-s + 1.37·53-s − 0.256·61-s − 0.977·67-s + 0.474·71-s + 0.702·73-s − 0.900·79-s + 0.878·83-s − 0.216·85-s − 0.635·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.52551102788331, −13.86520696417223, −13.49176381933956, −13.13608553494924, −12.58774185380268, −12.04346548131778, −11.69784317348999, −10.74197772280449, −10.50339639919068, −9.852385906430501, −9.560418512511243, −8.901236736375157, −8.529596153923905, −7.794701461033636, −6.965707771720678, −6.770974255763822, −6.163157424442823, −5.675895737916658, −5.003351880734074, −4.331610600715684, −3.630067736976559, −3.094205519612252, −2.517125924883022, −1.792146649560747, −0.8286008540299335, 0,
0.8286008540299335, 1.792146649560747, 2.517125924883022, 3.094205519612252, 3.630067736976559, 4.331610600715684, 5.003351880734074, 5.675895737916658, 6.163157424442823, 6.770974255763822, 6.965707771720678, 7.794701461033636, 8.529596153923905, 8.901236736375157, 9.560418512511243, 9.852385906430501, 10.50339639919068, 10.74197772280449, 11.69784317348999, 12.04346548131778, 12.58774185380268, 13.13608553494924, 13.49176381933956, 13.86520696417223, 14.52551102788331
