| L(s) = 1 | + 3-s + 2.64i·7-s + 9-s − 5.58i·11-s − 3.55·13-s + 1.58i·17-s − 4.58i·19-s + 2.64i·21-s + 7.84i·23-s + 27-s − 0.913i·29-s + 9.57·31-s − 5.58i·33-s + 6.92·37-s − 3.55·39-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.999i·7-s + 0.333·9-s − 1.68i·11-s − 0.987·13-s + 0.383i·17-s − 1.05i·19-s + 0.577i·21-s + 1.63i·23-s + 0.192·27-s − 0.169i·29-s + 1.71·31-s − 0.971i·33-s + 1.13·37-s − 0.569·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 + 0.200i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.294136336\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.294136336\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2.64iT - 7T^{2} \) |
| 11 | \( 1 + 5.58iT - 11T^{2} \) |
| 13 | \( 1 + 3.55T + 13T^{2} \) |
| 17 | \( 1 - 1.58iT - 17T^{2} \) |
| 19 | \( 1 + 4.58iT - 19T^{2} \) |
| 23 | \( 1 - 7.84iT - 23T^{2} \) |
| 29 | \( 1 + 0.913iT - 29T^{2} \) |
| 31 | \( 1 - 9.57T + 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 + 0.417T + 41T^{2} \) |
| 43 | \( 1 + 4.58T + 43T^{2} \) |
| 47 | \( 1 + 5.29iT - 47T^{2} \) |
| 53 | \( 1 - 6.20T + 53T^{2} \) |
| 59 | \( 1 - 9.16iT - 59T^{2} \) |
| 61 | \( 1 + 10.4iT - 61T^{2} \) |
| 67 | \( 1 - 4.58T + 67T^{2} \) |
| 71 | \( 1 - 0.723T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 + 18.1iT - 97T^{2} \) |
| 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
−8.313096695597053647149952376519, −7.73472349949969997217234094128, −6.81416341973299814060573830123, −6.01650859010379056141522591403, −5.39564108711982635246294064613, −4.58490957217793953349382921020, −3.47721694296628187205114623033, −2.86561973449232163829153997719, −2.11246413383116602698785094384, −0.74006494746940227302724778265,
0.887888490520634682819073053417, 2.10732647407149397017540521454, 2.74648525439782743112816425275, 3.96371094735335687639634792323, 4.48876115644814078891861639396, 5.07901429710839800850106951551, 6.47806093717871403817854966303, 6.87643133730457832436103540935, 7.76329859847928972539322714695, 7.984771138610333948932586374944
