L(s) = 1 | − 3.09·5-s + i·7-s + 0.646i·11-s − 4.37i·13-s + 0.295i·17-s + 5.29·19-s − 3.94·23-s + 4.58·25-s + 5.03·29-s + 9.16i·31-s − 3.09i·35-s − 2.55i·37-s + 10.4i·41-s + 1.63·43-s − 5.65·47-s + ⋯ |
L(s) = 1 | − 1.38·5-s + 0.377i·7-s + 0.194i·11-s − 1.21i·13-s + 0.0715i·17-s + 1.21·19-s − 0.823·23-s + 0.916·25-s + 0.934·29-s + 1.64i·31-s − 0.523i·35-s − 0.419i·37-s + 1.62i·41-s + 0.249·43-s − 0.825·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.639 + 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.639 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4630843679\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4630843679\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
good | 5 | \( 1 + 3.09T + 5T^{2} \) |
| 11 | \( 1 - 0.646iT - 11T^{2} \) |
| 13 | \( 1 + 4.37iT - 13T^{2} \) |
| 17 | \( 1 - 0.295iT - 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 + 3.94T + 23T^{2} \) |
| 29 | \( 1 - 5.03T + 29T^{2} \) |
| 31 | \( 1 - 9.16iT - 31T^{2} \) |
| 37 | \( 1 + 2.55iT - 37T^{2} \) |
| 41 | \( 1 - 10.4iT - 41T^{2} \) |
| 43 | \( 1 - 1.63T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + 8.64T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 2.55T + 67T^{2} \) |
| 71 | \( 1 - 1.11T + 71T^{2} \) |
| 73 | \( 1 + 9.58T + 73T^{2} \) |
| 79 | \( 1 + 16.7iT - 79T^{2} \) |
| 83 | \( 1 + 8.50iT - 83T^{2} \) |
| 89 | \( 1 + 10.4iT - 89T^{2} \) |
| 97 | \( 1 + 2.41T + 97T^{2} \) |
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
−8.032699405381475211855996619825, −7.68588582078155981718576607284, −6.82773391382860713193043230845, −5.94745640703027056928136861210, −5.05485832815298037933186914159, −4.44136101582430613400638149554, −3.30984088760659586656983702990, −3.00833403766688459668699304043, −1.42702576049018909315258581933, −0.15906754049462487837373373401,
1.06934599763782384526145005732, 2.40804446558771210072405662354, 3.53541666551531831774256477319, 4.06817433445988265856862522977, 4.73699098254377015498527578635, 5.76672785841780206376177697387, 6.70392570038257813670615959632, 7.31395821233384324125278896188, 7.943453840062460838014223094976, 8.515891503369569878324573355038