L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 1.23·11-s + 12-s − 1.23·13-s + 14-s + 15-s + 16-s + 6.47·17-s − 18-s + 4.47·19-s + 20-s − 21-s − 1.23·22-s + 23-s − 24-s + 25-s + 1.23·26-s + 27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.372·11-s + 0.288·12-s − 0.342·13-s + 0.267·14-s + 0.258·15-s + 0.250·16-s + 1.56·17-s − 0.235·18-s + 1.02·19-s + 0.223·20-s − 0.218·21-s − 0.263·22-s + 0.208·23-s − 0.204·24-s + 0.200·25-s + 0.242·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.044539160\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.044539160\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 - 1.23T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 - 8.47T + 37T^{2} \) |
| 41 | \( 1 + 0.472T + 41T^{2} \) |
| 43 | \( 1 + 5.23T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 + 3.70T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 8.47T + 83T^{2} \) |
| 89 | \( 1 - 2.76T + 89T^{2} \) |
| 97 | \( 1 + 7.70T + 97T^{2} \) |
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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
−8.287588536890770274413637304999, −7.56708274635814454890847957683, −7.13880802825351487450414238354, −6.12974142443436564880072227242, −5.57104383784098561093169610635, −4.53114614787236694140449008053, −3.38581729956274980599513608993, −2.90136211314482545414220723143, −1.78898968673255261136769108942, −0.891540137263366751752946541433,
0.891540137263366751752946541433, 1.78898968673255261136769108942, 2.90136211314482545414220723143, 3.38581729956274980599513608993, 4.53114614787236694140449008053, 5.57104383784098561093169610635, 6.12974142443436564880072227242, 7.13880802825351487450414238354, 7.56708274635814454890847957683, 8.287588536890770274413637304999