| L(s) = 1 | + 5-s + 4·11-s + 2·13-s + 2·17-s + 4·19-s + 4·23-s + 25-s + 2·29-s − 8·31-s + 6·37-s − 6·41-s + 8·43-s − 4·47-s − 6·53-s + 4·55-s + 4·59-s + 2·61-s + 2·65-s − 8·67-s + 6·73-s + 16·83-s + 2·85-s − 6·89-s + 4·95-s + 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.986·37-s − 0.937·41-s + 1.21·43-s − 0.583·47-s − 0.824·53-s + 0.539·55-s + 0.520·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s + 0.702·73-s + 1.75·83-s + 0.216·85-s − 0.635·89-s + 0.410·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.524490948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.524490948\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.73802754081346, −14.44359743283061, −13.98784164950440, −13.32219125108185, −12.95187841888325, −12.26766472604867, −11.78567750930786, −11.22357386459491, −10.79862094750859, −10.07611564684429, −9.494695733892280, −9.131709774722811, −8.644534324359050, −7.822217156979086, −7.345430388461787, −6.645311368613738, −6.204565216899671, −5.553417691938207, −4.999646167073406, −4.249874490593700, −3.522064829754538, −3.104690799660728, −2.103662175729141, −1.371356847318043, −0.7724438171067429,
0.7724438171067429, 1.371356847318043, 2.103662175729141, 3.104690799660728, 3.522064829754538, 4.249874490593700, 4.999646167073406, 5.553417691938207, 6.204565216899671, 6.645311368613738, 7.345430388461787, 7.822217156979086, 8.644534324359050, 9.131709774722811, 9.494695733892280, 10.07611564684429, 10.79862094750859, 11.22357386459491, 11.78567750930786, 12.26766472604867, 12.95187841888325, 13.32219125108185, 13.98784164950440, 14.44359743283061, 14.73802754081346
