| L(s) = 1 | + 5-s + 6·11-s + 2·17-s − 2·19-s + 6·23-s + 25-s + 6·29-s + 8·31-s − 10·37-s + 4·41-s + 8·47-s − 7·49-s + 4·53-s + 6·55-s − 10·59-s + 10·61-s − 8·67-s + 4·71-s − 2·73-s + 16·79-s + 2·85-s − 2·95-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.80·11-s + 0.485·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s − 1.64·37-s + 0.624·41-s + 1.16·47-s − 49-s + 0.549·53-s + 0.809·55-s − 1.30·59-s + 1.28·61-s − 0.977·67-s + 0.474·71-s − 0.234·73-s + 1.80·79-s + 0.216·85-s − 0.205·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.928409950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.928409950\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.29339437370987, −13.71599525100302, −13.55989444940581, −12.63657846483549, −12.20143569792429, −11.94682620016307, −11.24782630463265, −10.68885868327080, −10.27001364086844, −9.484586141076773, −9.330666668010182, −8.534875643138117, −8.363334643961922, −7.369509685458457, −6.867280671343378, −6.450046501849809, −5.982928132453553, −5.218401660012913, −4.644039761377877, −4.095850070272240, −3.386891995082693, −2.837273536640425, −1.996732602711521, −1.258627211488363, −0.7610565762242350,
0.7610565762242350, 1.258627211488363, 1.996732602711521, 2.837273536640425, 3.386891995082693, 4.095850070272240, 4.644039761377877, 5.218401660012913, 5.982928132453553, 6.450046501849809, 6.867280671343378, 7.369509685458457, 8.363334643961922, 8.534875643138117, 9.330666668010182, 9.484586141076773, 10.27001364086844, 10.68885868327080, 11.24782630463265, 11.94682620016307, 12.20143569792429, 12.63657846483549, 13.55989444940581, 13.71599525100302, 14.29339437370987
