| L(s) = 1 | − 2·4-s − 3·9-s + 11-s − 5·13-s + 4·16-s − 5·17-s − 5·19-s − 4·29-s − 10·31-s + 6·36-s + 5·37-s − 5·41-s − 10·43-s − 2·44-s − 10·47-s + 10·52-s − 5·53-s − 10·59-s + 5·61-s − 8·64-s + 5·67-s + 10·68-s + 3·71-s + 5·73-s + 10·76-s + 4·79-s + 9·81-s + ⋯ |
| L(s) = 1 | − 4-s − 9-s + 0.301·11-s − 1.38·13-s + 16-s − 1.21·17-s − 1.14·19-s − 0.742·29-s − 1.79·31-s + 36-s + 0.821·37-s − 0.780·41-s − 1.52·43-s − 0.301·44-s − 1.45·47-s + 1.38·52-s − 0.686·53-s − 1.30·59-s + 0.640·61-s − 64-s + 0.610·67-s + 1.21·68-s + 0.356·71-s + 0.585·73-s + 1.14·76-s + 0.450·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−16.91222648529839, −16.53239856428240, −15.41455075889520, −14.89999493818156, −14.55525407125700, −14.10021894922854, −13.25318284018864, −12.95265226966762, −12.38962269679311, −11.60088026257895, −11.14521411674947, −10.46955829394947, −9.694634396354839, −9.276366257030594, −8.722836161320256, −8.173344200206868, −7.533347401207620, −6.660465492170884, −6.139817632574795, −5.098730645978104, −4.979610581748358, −4.040377436329884, −3.418091306667412, −2.465834691523017, −1.719735668092142, 0, 0,
1.719735668092142, 2.465834691523017, 3.418091306667412, 4.040377436329884, 4.979610581748358, 5.098730645978104, 6.139817632574795, 6.660465492170884, 7.533347401207620, 8.173344200206868, 8.722836161320256, 9.276366257030594, 9.694634396354839, 10.46955829394947, 11.14521411674947, 11.60088026257895, 12.38962269679311, 12.95265226966762, 13.25318284018864, 14.10021894922854, 14.55525407125700, 14.89999493818156, 15.41455075889520, 16.53239856428240, 16.91222648529839
