| L(s) = 1 | + (−1.16 + 1.58i)2-s + (−1.07 + 0.203i)3-s + (−0.551 − 1.78i)4-s + (1.88 − 1.19i)5-s + (0.935 − 1.94i)6-s + (−1.33 + 2.28i)7-s + (−0.239 − 0.0837i)8-s + (−1.67 + 0.656i)9-s + (−0.315 + 4.38i)10-s + (−0.791 + 2.01i)11-s + (0.958 + 1.81i)12-s + (−6.58 − 0.742i)13-s + (−2.05 − 4.78i)14-s + (−1.79 + 1.67i)15-s + (3.50 − 2.38i)16-s + (−1.13 − 0.0422i)17-s + ⋯ |
| L(s) = 1 | + (−0.826 + 1.11i)2-s + (−0.621 + 0.117i)3-s + (−0.275 − 0.893i)4-s + (0.845 − 0.534i)5-s + (0.382 − 0.793i)6-s + (−0.505 + 0.862i)7-s + (−0.0846 − 0.0296i)8-s + (−0.557 + 0.218i)9-s + (−0.0998 + 1.38i)10-s + (−0.238 + 0.608i)11-s + (0.276 + 0.523i)12-s + (−1.82 − 0.205i)13-s + (−0.548 − 1.27i)14-s + (−0.462 + 0.431i)15-s + (0.876 − 0.597i)16-s + (−0.274 − 0.0102i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0871249 - 0.206247i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0871249 - 0.206247i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.88 + 1.19i)T \) |
| 7 | \( 1 + (1.33 - 2.28i)T \) |
| good | 2 | \( 1 + (1.16 - 1.58i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (1.07 - 0.203i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (0.791 - 2.01i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (6.58 + 0.742i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (1.13 + 0.0422i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (-2.02 + 3.50i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.181 - 4.83i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (9.97 - 2.27i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (1.41 - 0.817i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.86 - 2.57i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-0.233 - 0.484i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (1.77 + 5.06i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-5.98 - 4.41i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (-6.19 - 3.27i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.660 - 8.81i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (2.54 - 8.24i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (6.92 + 1.85i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (0.195 - 0.855i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-9.10 + 6.71i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-12.8 - 7.41i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.49 - 13.2i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-3.58 - 9.12i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (1.88 + 1.88i)T + 97iT^{2} \) |
| 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
−12.51214910459869111640976058630, −11.90699552647187165660732539264, −10.38824206455646623919785475031, −9.384826566312798635331926935545, −9.066232447232790379268459550516, −7.66901372440583185742804354892, −6.76914466458988816254306068181, −5.46315582114672151907043704254, −5.27642885483953131740123440710, −2.53874167732718151483497883438,
0.22914644735805300957958691557, 2.19785216347369371075920017751, 3.40105134729273854034933996482, 5.38864795200425122436629372632, 6.42224421543168947564435899002, 7.57110932914353113047735228128, 9.071872075962670743060929788425, 9.845797700232654691480738523733, 10.52076980163706219323345303545, 11.23730587562434770710312174803
