| L(s) = 1 | + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−3.5 + 6.06i)5-s + 7.99·8-s + (−7 − 12.1i)10-s + (17.5 + 30.3i)11-s − 66·13-s + (−8 + 13.8i)16-s + (−29.5 − 51.0i)17-s + (68.5 − 118. i)19-s + 28·20-s − 70·22-s + (−3.5 + 6.06i)23-s + (38 + 65.8i)25-s + (66 − 114. i)26-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.313 + 0.542i)5-s + 0.353·8-s + (−0.221 − 0.383i)10-s + (0.479 + 0.830i)11-s − 1.40·13-s + (−0.125 + 0.216i)16-s + (−0.420 − 0.728i)17-s + (0.827 − 1.43i)19-s + 0.313·20-s − 0.678·22-s + (−0.0317 + 0.0549i)23-s + (0.303 + 0.526i)25-s + (0.497 − 0.862i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.022900609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.022900609\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 - 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (3.5 - 6.06i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-17.5 - 30.3i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 66T + 2.19e3T^{2} \) |
| 17 | \( 1 + (29.5 + 51.0i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-68.5 + 118. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (3.5 - 6.06i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 106T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-37.5 - 64.9i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (5.5 - 9.52i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 498T + 6.89e4T^{2} \) |
| 43 | \( 1 - 260T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-85.5 + 148. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (208.5 + 361. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-8.5 - 14.7i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-25.5 + 44.1i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (219.5 + 380. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 784T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-147.5 - 255. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-247.5 + 428. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 932T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-436.5 + 756. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 290T + 9.12e5T^{2} \) |
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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
−9.520398906676197926588163413478, −9.076260019416269867917586082603, −7.78204159044455625372466350830, −7.07178880730073154365133204922, −6.75390374056931334616334687400, −5.21205112577994915283732849663, −4.67988562440756188989188847818, −3.25449409154516855377114838133, −2.07252921039306952139782110772, −0.40338328084342301524641908582,
0.832043311305833913523232234021, 2.03693154474428314659159210349, 3.32198324117737817605549476503, 4.21354458573971250352090719392, 5.23451480780146545295883837705, 6.30000082153689471637851609023, 7.51927821940859444470299721761, 8.187452947395834703355436028831, 8.997042314847306825319242943007, 9.783844137183922726823207024269
