L(s) = 1 | − 3-s − 3.20·7-s + 9-s + 0.359·11-s + 5.92·13-s + 4.79·17-s + 4.51·19-s + 3.20·21-s + 23-s − 27-s − 7.36·29-s − 6.26·31-s − 0.359·33-s + 1.20·37-s − 5.92·39-s + 1.85·41-s + 10.8·43-s + 7.51·47-s + 3.30·49-s − 4.79·51-s + 2.69·53-s − 4.51·57-s − 7.00·59-s + 0.152·61-s − 3.20·63-s − 5.77·67-s − 69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.21·7-s + 0.333·9-s + 0.108·11-s + 1.64·13-s + 1.16·17-s + 1.03·19-s + 0.700·21-s + 0.208·23-s − 0.192·27-s − 1.36·29-s − 1.12·31-s − 0.0625·33-s + 0.198·37-s − 0.949·39-s + 0.289·41-s + 1.65·43-s + 1.09·47-s + 0.471·49-s − 0.671·51-s + 0.370·53-s − 0.597·57-s − 0.911·59-s + 0.0195·61-s − 0.404·63-s − 0.705·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.517428438\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.517428438\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 3.20T + 7T^{2} \) |
| 11 | \( 1 - 0.359T + 11T^{2} \) |
| 13 | \( 1 - 5.92T + 13T^{2} \) |
| 17 | \( 1 - 4.79T + 17T^{2} \) |
| 19 | \( 1 - 4.51T + 19T^{2} \) |
| 29 | \( 1 + 7.36T + 29T^{2} \) |
| 31 | \( 1 + 6.26T + 31T^{2} \) |
| 37 | \( 1 - 1.20T + 37T^{2} \) |
| 41 | \( 1 - 1.85T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 - 7.51T + 47T^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 + 7.00T + 59T^{2} \) |
| 61 | \( 1 - 0.152T + 61T^{2} \) |
| 67 | \( 1 + 5.77T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 5.28T + 73T^{2} \) |
| 79 | \( 1 + 0.454T + 79T^{2} \) |
| 83 | \( 1 - 1.01T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 + 4.57T + 97T^{2} \) |
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
−7.64226548631992034772311964089, −7.36579755918461285213913153403, −6.38500053863756365545637266480, −5.79014138343836652222019499849, −5.50815836892341751659261795579, −4.20083694455674769935007698522, −3.57291650259892844675201130740, −2.97926508025676688173703212560, −1.56971039333933007681421752110, −0.68540088867472909955337346897,
0.68540088867472909955337346897, 1.56971039333933007681421752110, 2.97926508025676688173703212560, 3.57291650259892844675201130740, 4.20083694455674769935007698522, 5.50815836892341751659261795579, 5.79014138343836652222019499849, 6.38500053863756365545637266480, 7.36579755918461285213913153403, 7.64226548631992034772311964089