| L(s) = 1 | + (−0.661 + 0.661i)5-s + (−2.79 − 0.750i)7-s + (4.25 − 1.13i)11-s + (1.70 + 3.17i)13-s + (−0.870 − 1.50i)17-s + (−0.589 + 2.20i)19-s + (1.96 − 3.40i)23-s + 4.12i·25-s + (4.62 + 2.67i)29-s + (4.82 + 4.82i)31-s + (2.34 − 1.35i)35-s + (2.86 + 10.6i)37-s + (−1.04 − 3.88i)41-s + (6.62 − 3.82i)43-s + (6.84 + 6.84i)47-s + ⋯ |
| L(s) = 1 | + (−0.295 + 0.295i)5-s + (−1.05 − 0.283i)7-s + (1.28 − 0.343i)11-s + (0.471 + 0.881i)13-s + (−0.211 − 0.365i)17-s + (−0.135 + 0.505i)19-s + (0.409 − 0.709i)23-s + 0.825i·25-s + (0.859 + 0.495i)29-s + (0.867 + 0.867i)31-s + (0.396 − 0.229i)35-s + (0.470 + 1.75i)37-s + (−0.162 − 0.606i)41-s + (1.00 − 0.583i)43-s + (0.998 + 0.998i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29579 + 0.463579i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29579 + 0.463579i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-1.70 - 3.17i)T \) |
| good | 5 | \( 1 + (0.661 - 0.661i)T - 5iT^{2} \) |
| 7 | \( 1 + (2.79 + 0.750i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-4.25 + 1.13i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (0.870 + 1.50i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.589 - 2.20i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.96 + 3.40i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.62 - 2.67i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.82 - 4.82i)T + 31iT^{2} \) |
| 37 | \( 1 + (-2.86 - 10.6i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (1.04 + 3.88i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-6.62 + 3.82i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.84 - 6.84i)T + 47iT^{2} \) |
| 53 | \( 1 - 8.89iT - 53T^{2} \) |
| 59 | \( 1 + (-1.63 + 6.08i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (2.39 + 4.15i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.52 - 2.55i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (2.88 + 0.771i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (0.315 - 0.315i)T - 73iT^{2} \) |
| 79 | \( 1 + 1.75T + 79T^{2} \) |
| 83 | \( 1 + (-9.41 + 9.41i)T - 83iT^{2} \) |
| 89 | \( 1 + (-12.6 + 3.40i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (1.85 - 6.91i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−10.17563475184700899862078287199, −9.170774016672123199752214687705, −8.758035353419359508527261193952, −7.45687011853990655826260151259, −6.54531133185340228053308355867, −6.26996205615264666348034439590, −4.67514833716136737959276498930, −3.76251647459368676739284365776, −2.93253322406729663656339954880, −1.19263927847696726227444447487,
0.792403382045317190891647891596, 2.52937589748004598775113026257, 3.67625204633903732110027962427, 4.47607695828676785598426958631, 5.86267878411726954592086857836, 6.41362148428183627606316333769, 7.40807777249822326676154708612, 8.422871698388466896229442709095, 9.172119416999286594953829310530, 9.833533250900768816930129656504
