L(s) = 1 | + 2.61i·5-s + (2.61 + 0.414i)7-s − 2i·11-s − 4.77i·13-s + 3.06i·17-s − 4.14·19-s + 7.65i·23-s − 1.82·25-s + 3.65·29-s − 3.06·31-s + (−1.08 + 6.82i)35-s + 7.65·37-s + 9.55i·41-s − 3.65i·43-s + 7.39·47-s + ⋯ |
L(s) = 1 | + 1.16i·5-s + (0.987 + 0.156i)7-s − 0.603i·11-s − 1.32i·13-s + 0.742i·17-s − 0.950·19-s + 1.59i·23-s − 0.365·25-s + 0.679·29-s − 0.549·31-s + (−0.182 + 1.15i)35-s + 1.25·37-s + 1.49i·41-s − 0.557i·43-s + 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.957532843\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.957532843\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.61 - 0.414i)T \) |
good | 5 | \( 1 - 2.61iT - 5T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 4.77iT - 13T^{2} \) |
| 17 | \( 1 - 3.06iT - 17T^{2} \) |
| 19 | \( 1 + 4.14T + 19T^{2} \) |
| 23 | \( 1 - 7.65iT - 23T^{2} \) |
| 29 | \( 1 - 3.65T + 29T^{2} \) |
| 31 | \( 1 + 3.06T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 - 9.55iT - 41T^{2} \) |
| 43 | \( 1 + 3.65iT - 43T^{2} \) |
| 47 | \( 1 - 7.39T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 8.47T + 59T^{2} \) |
| 61 | \( 1 - 2.61iT - 61T^{2} \) |
| 67 | \( 1 - 15.6iT - 67T^{2} \) |
| 71 | \( 1 + 8.82iT - 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 + 12.8iT - 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 2.16iT - 89T^{2} \) |
| 97 | \( 1 - 13.5iT - 97T^{2} \) |
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
−8.426198496534133126833900398467, −7.84022639513046865422999068110, −7.32065760731317574238590845692, −6.23217715516393226993223780725, −5.82354284519752936766848139615, −4.91788974346155340617038392393, −3.90006631079592398353871054977, −3.11056167959962020054233720459, −2.33183834053790796068972164540, −1.15085227969323348129966864271,
0.61508377128322386821236700062, 1.73413510574749569023784681742, 2.44024087931290368483116396851, 4.04715989928305517957348234697, 4.62963415310816937686518134793, 4.91727288815079419085433612547, 6.07584354200976477692615062070, 6.85301172845406983499135465512, 7.63679526493458604803272494826, 8.387021367309161646986563700660