| L(s) = 1 | − 3·5-s + 13-s − 3·17-s − 4·19-s + 4·25-s − 9·29-s − 4·31-s − 37-s + 6·41-s − 8·43-s + 12·47-s + 6·53-s + 61-s − 3·65-s + 4·67-s − 12·71-s − 11·73-s + 16·79-s + 12·83-s + 9·85-s − 3·89-s + 12·95-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.277·13-s − 0.727·17-s − 0.917·19-s + 4/5·25-s − 1.67·29-s − 0.718·31-s − 0.164·37-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + 0.824·53-s + 0.128·61-s − 0.372·65-s + 0.488·67-s − 1.42·71-s − 1.28·73-s + 1.80·79-s + 1.31·83-s + 0.976·85-s − 0.317·89-s + 1.23·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.73439175357181, −13.98746530232753, −13.37752393126114, −13.00550325963403, −12.42582029457649, −11.97607731071981, −11.43833062443200, −10.94465824688734, −10.71905876412433, −9.946168986402910, −9.232694456933365, −8.763529691122478, −8.406115814535576, −7.640795327812410, −7.359966702313731, −6.817776082014724, −6.072185482409929, −5.586669101994147, −4.777088069867199, −4.223556331600563, −3.799724650219544, −3.291858904662745, −2.358180125797893, −1.801230555516983, −0.6984545628888183, 0,
0.6984545628888183, 1.801230555516983, 2.358180125797893, 3.291858904662745, 3.799724650219544, 4.223556331600563, 4.777088069867199, 5.586669101994147, 6.072185482409929, 6.817776082014724, 7.359966702313731, 7.640795327812410, 8.406115814535576, 8.763529691122478, 9.232694456933365, 9.946168986402910, 10.71905876412433, 10.94465824688734, 11.43833062443200, 11.97607731071981, 12.42582029457649, 13.00550325963403, 13.37752393126114, 13.98746530232753, 14.73439175357181
