| L(s) = 1 | − 4·5-s − 11-s + 4·13-s − 6·17-s + 2·19-s + 11·25-s + 10·29-s + 8·31-s + 6·37-s + 2·41-s − 12·43-s − 8·47-s + 6·53-s + 4·55-s − 14·59-s − 16·65-s + 8·67-s + 8·71-s + 10·73-s − 8·79-s − 10·83-s + 24·85-s + 2·89-s − 8·95-s − 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.301·11-s + 1.10·13-s − 1.45·17-s + 0.458·19-s + 11/5·25-s + 1.85·29-s + 1.43·31-s + 0.986·37-s + 0.312·41-s − 1.82·43-s − 1.16·47-s + 0.824·53-s + 0.539·55-s − 1.82·59-s − 1.98·65-s + 0.977·67-s + 0.949·71-s + 1.17·73-s − 0.900·79-s − 1.09·83-s + 2.60·85-s + 0.211·89-s − 0.820·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.619011143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.619011143\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−12.55441944831349, −12.13178191579273, −11.67485396558674, −11.26491419007539, −11.07462510638985, −10.47589781347293, −9.997027564972781, −9.405293408480338, −8.744771418689229, −8.347128785714895, −8.155212072144702, −7.752937441861291, −6.886372278608261, −6.724869542744746, −6.303507670389597, −5.534453146350431, −4.741820546511093, −4.569531011232850, −4.121007027530376, −3.447659143347742, −3.047394629891292, −2.562884972228944, −1.642126074663396, −0.8960798476655564, −0.4324773846569843,
0.4324773846569843, 0.8960798476655564, 1.642126074663396, 2.562884972228944, 3.047394629891292, 3.447659143347742, 4.121007027530376, 4.569531011232850, 4.741820546511093, 5.534453146350431, 6.303507670389597, 6.724869542744746, 6.886372278608261, 7.752937441861291, 8.155212072144702, 8.347128785714895, 8.744771418689229, 9.405293408480338, 9.997027564972781, 10.47589781347293, 11.07462510638985, 11.26491419007539, 11.67485396558674, 12.13178191579273, 12.55441944831349
