| L(s) = 1 | + 0.765·5-s + 4i·11-s − 4.90i·13-s − 5.54·17-s + 7.39i·19-s − 5.65i·23-s − 4.41·25-s + 1.65i·29-s + 4.32i·31-s + 8.24·37-s + 1.84·41-s + 1.65·43-s − 7.39·47-s − 8.24i·53-s + 3.06i·55-s + ⋯ |
| L(s) = 1 | + 0.342·5-s + 1.20i·11-s − 1.36i·13-s − 1.34·17-s + 1.69i·19-s − 1.17i·23-s − 0.882·25-s + 0.307i·29-s + 0.777i·31-s + 1.35·37-s + 0.288·41-s + 0.252·43-s − 1.07·47-s − 1.13i·53-s + 0.412i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.716 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5486112421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5486112421\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 - 0.765T + 5T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 4.90iT - 13T^{2} \) |
| 17 | \( 1 + 5.54T + 17T^{2} \) |
| 19 | \( 1 - 7.39iT - 19T^{2} \) |
| 23 | \( 1 + 5.65iT - 23T^{2} \) |
| 29 | \( 1 - 1.65iT - 29T^{2} \) |
| 31 | \( 1 - 4.32iT - 31T^{2} \) |
| 37 | \( 1 - 8.24T + 37T^{2} \) |
| 41 | \( 1 - 1.84T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 + 8.24iT - 53T^{2} \) |
| 59 | \( 1 - 4.32T + 59T^{2} \) |
| 61 | \( 1 + 14.0iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 13.6iT - 71T^{2} \) |
| 73 | \( 1 + 3.82iT - 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 + 18.6T + 89T^{2} \) |
| 97 | \( 1 - 3.82iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84108678023722453330283970827, −6.90989904200010690648442299187, −6.29250956458355249635375932665, −5.60797997721995005939443140368, −4.79229703305038589750048142912, −4.16570944595245104610929094830, −3.21420131540002596796067763709, −2.28766363385380413799740536679, −1.58202901099347980069768496485, −0.13035398564816548691145416607,
1.16183852611262260422672050840, 2.26511211861947550927950154693, 2.87473626367823894042400612120, 4.11406524946636095291062330737, 4.41866607850170742545883896566, 5.56981166566423410286380674783, 6.05796782680585196920463114187, 6.82403264426271593441881706796, 7.36732724155986929881621447456, 8.331423237768419128629770898424
