L(s) = 1 | − 1.41i·5-s − 4.24·7-s − 6i·11-s − 5.65i·13-s + 6·17-s − 4i·19-s − 2.82·23-s + 2.99·25-s + 1.41i·29-s + 1.41·31-s + 6i·35-s − 8.48i·37-s − 2·41-s + 2.82·47-s + 10.9·49-s + ⋯ |
L(s) = 1 | − 0.632i·5-s − 1.60·7-s − 1.80i·11-s − 1.56i·13-s + 1.45·17-s − 0.917i·19-s − 0.589·23-s + 0.599·25-s + 0.262i·29-s + 0.254·31-s + 1.01i·35-s − 1.39i·37-s − 0.312·41-s + 0.412·47-s + 1.57·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.075581905\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.075581905\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 8.48iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 9.89iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 8.48iT - 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 12.7T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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
−8.003747365264396275336175339335, −7.31058776131590515537392188278, −6.29386412975003772955155784112, −5.73833781821420961057648277166, −5.30440109684524711458054546474, −3.98995014971708322824379983856, −3.11147592997510932958486347741, −2.91550467341414087106561863396, −0.958441861973466185745875123418, −0.36194910877997420341899303686,
1.52334033178258868845068614468, 2.47532082875647252763627035496, 3.38095059180083316229412886294, 4.04667328153735772429132662030, 4.92312212662441169763262122129, 6.01703623059964665732500433770, 6.60826713033733301352882858055, 7.07731145002112052364342680363, 7.74475827291824036185551198009, 8.798159900511648928386303106462