| L(s) = 1 | + 3-s − 0.331·5-s − 3.08·7-s + 9-s − 3.69·11-s + 4.64·13-s − 0.331·15-s + 6.52·17-s − 0.867·19-s − 3.08·21-s − 4·23-s − 4.88·25-s + 27-s − 4.89·29-s − 6.14·31-s − 3.69·33-s + 1.02·35-s − 3.64·37-s + 4.64·39-s + 3.92·41-s + 3.92·43-s − 0.331·45-s + 1.65·47-s + 2.50·49-s + 6.52·51-s + 0.564·53-s + 1.22·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.148·5-s − 1.16·7-s + 0.333·9-s − 1.11·11-s + 1.28·13-s − 0.0856·15-s + 1.58·17-s − 0.198·19-s − 0.672·21-s − 0.834·23-s − 0.977·25-s + 0.192·27-s − 0.908·29-s − 1.10·31-s − 0.643·33-s + 0.172·35-s − 0.599·37-s + 0.743·39-s + 0.613·41-s + 0.599·43-s − 0.0494·45-s + 0.241·47-s + 0.357·49-s + 0.913·51-s + 0.0775·53-s + 0.165·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 + 0.331T + 5T^{2} \) |
| 7 | \( 1 + 3.08T + 7T^{2} \) |
| 11 | \( 1 + 3.69T + 11T^{2} \) |
| 13 | \( 1 - 4.64T + 13T^{2} \) |
| 17 | \( 1 - 6.52T + 17T^{2} \) |
| 19 | \( 1 + 0.867T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4.89T + 29T^{2} \) |
| 31 | \( 1 + 6.14T + 31T^{2} \) |
| 37 | \( 1 + 3.64T + 37T^{2} \) |
| 41 | \( 1 - 3.92T + 41T^{2} \) |
| 43 | \( 1 - 3.92T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 - 0.564T + 53T^{2} \) |
| 59 | \( 1 + 6.59T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 + 7.49T + 71T^{2} \) |
| 73 | \( 1 - 5.62T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 - 9.35T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 + 17.0T + 97T^{2} \) |
| 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
−8.143334228419237591586128826342, −7.78945515937108546808995503682, −6.93213738788903464184043134385, −5.87215642705597857775916473196, −5.56725283300436171416629233356, −4.07437701849818534347962253892, −3.51679981096337826400463689735, −2.77668156654658852982934971027, −1.57284789748032231902904687974, 0,
1.57284789748032231902904687974, 2.77668156654658852982934971027, 3.51679981096337826400463689735, 4.07437701849818534347962253892, 5.56725283300436171416629233356, 5.87215642705597857775916473196, 6.93213738788903464184043134385, 7.78945515937108546808995503682, 8.143334228419237591586128826342
