L(s) = 1 | + (1.5 − 2.59i)5-s + (−2.5 − 0.866i)7-s + (4.5 − 2.59i)11-s + (3 + 1.73i)13-s + 6·17-s − 1.73i·19-s + (4.5 + 2.59i)23-s + (−2 − 3.46i)25-s + (−9 + 5.19i)29-s + (4.5 + 2.59i)31-s + (−6 + 5.19i)35-s + 37-s + (−1.5 + 2.59i)41-s + (−5 − 8.66i)43-s + (−3 − 5.19i)47-s + ⋯ |
L(s) = 1 | + (0.670 − 1.16i)5-s + (−0.944 − 0.327i)7-s + (1.35 − 0.783i)11-s + (0.832 + 0.480i)13-s + 1.45·17-s − 0.397i·19-s + (0.938 + 0.541i)23-s + (−0.400 − 0.692i)25-s + (−1.67 + 0.964i)29-s + (0.808 + 0.466i)31-s + (−1.01 + 0.878i)35-s + 0.164·37-s + (−0.234 + 0.405i)41-s + (−0.762 − 1.32i)43-s + (−0.437 − 0.757i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.151136066\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.151136066\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.5 + 2.59i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3 - 1.73i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 + (-4.5 - 2.59i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (9 - 5.19i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.5 - 2.59i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-12 + 6.92i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 5.19iT - 71T^{2} \) |
| 73 | \( 1 - 3.46iT - 73T^{2} \) |
| 79 | \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.971169305644872959816237085069, −8.416836886736875482224823691529, −7.15985897898664530680364618036, −6.49831050372623773386126757140, −5.69327221825384313571380470349, −5.06113459125429457780400616757, −3.75832181604404260756758000886, −3.35588428468028598045394777770, −1.59385207033353707983056220150, −0.885021576774499407782085664412,
1.28318635902816524425711520816, 2.53168421958939540825964796948, 3.33266659538549056966651776267, 4.07749960917803374156680789587, 5.54717702203720572825408905269, 6.14803186350318474106581110272, 6.70316113740652510031768526515, 7.44155411449298065825338825650, 8.448331693819944350159991563158, 9.507470013495837683527479355557