| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s + 11-s + 6·13-s + 2·15-s − 2·17-s − 8·19-s + 21-s + 4·23-s − 25-s − 27-s − 2·29-s + 8·31-s − 33-s + 2·35-s − 6·37-s − 6·39-s − 2·41-s + 8·43-s − 2·45-s + 4·47-s + 49-s + 2·51-s − 2·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s + 0.516·15-s − 0.485·17-s − 1.83·19-s + 0.218·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.174·33-s + 0.338·35-s − 0.986·37-s − 0.960·39-s − 0.312·41-s + 1.21·43-s − 0.298·45-s + 0.583·47-s + 1/7·49-s + 0.280·51-s − 0.274·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−16.32319535807932, −15.74971236864546, −15.23297130880236, −15.11921488883406, −13.88416657606029, −13.62361771807136, −12.90295052954974, −12.34485926986902, −11.91809664617286, −11.14005958841486, −10.75738902494129, −10.47688704600195, −9.273385437864477, −8.976228884313233, −8.212765872368441, −7.760591591714704, −6.769005646706982, −6.423527265282056, −5.931942291099760, −4.937945076651696, −4.213204824490428, −3.837141043925026, −3.022528499874497, −1.936451465172219, −0.9618170965035186, 0,
0.9618170965035186, 1.936451465172219, 3.022528499874497, 3.837141043925026, 4.213204824490428, 4.937945076651696, 5.931942291099760, 6.423527265282056, 6.769005646706982, 7.760591591714704, 8.212765872368441, 8.976228884313233, 9.273385437864477, 10.47688704600195, 10.75738902494129, 11.14005958841486, 11.91809664617286, 12.34485926986902, 12.90295052954974, 13.62361771807136, 13.88416657606029, 15.11921488883406, 15.23297130880236, 15.74971236864546, 16.32319535807932
