L(s) = 1 | + 0.823i·2-s + (−1.33 + 2.30i)3-s + 1.32·4-s + (2.73 + 1.58i)5-s + (−1.89 − 1.09i)6-s + 2.73i·8-s + (−2.03 − 3.53i)9-s + (−1.30 + 2.25i)10-s + (−5.14 − 2.97i)11-s + (−1.75 + 3.04i)12-s + (−0.0766 + 3.60i)13-s + (−7.28 + 4.20i)15-s + 0.390·16-s + 2.69·17-s + (2.90 − 1.67i)18-s + (−1.69 + 0.978i)19-s + ⋯ |
L(s) = 1 | + 0.582i·2-s + (−0.767 + 1.33i)3-s + 0.660·4-s + (1.22 + 0.707i)5-s + (−0.774 − 0.447i)6-s + 0.967i·8-s + (−0.679 − 1.17i)9-s + (−0.411 + 0.713i)10-s + (−1.55 − 0.895i)11-s + (−0.507 + 0.879i)12-s + (−0.0212 + 0.999i)13-s + (−1.88 + 1.08i)15-s + 0.0976·16-s + 0.654·17-s + (0.685 − 0.395i)18-s + (−0.388 + 0.224i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.121132 + 1.50312i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.121132 + 1.50312i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (0.0766 - 3.60i)T \) |
good | 2 | \( 1 - 0.823iT - 2T^{2} \) |
| 3 | \( 1 + (1.33 - 2.30i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.73 - 1.58i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (5.14 + 2.97i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 2.69T + 17T^{2} \) |
| 19 | \( 1 + (1.69 - 0.978i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 2.72T + 23T^{2} \) |
| 29 | \( 1 + (-2.99 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.997 - 0.575i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.50iT - 37T^{2} \) |
| 41 | \( 1 + (3.23 - 1.86i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.49 + 6.05i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.394 + 0.228i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.199 + 0.345i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 4.80iT - 59T^{2} \) |
| 61 | \( 1 + (-0.578 - 1.00i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.43 - 3.13i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.90 - 2.25i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.19 + 4.15i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.95 + 6.85i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.19iT - 83T^{2} \) |
| 89 | \( 1 - 3.56iT - 89T^{2} \) |
| 97 | \( 1 + (2.96 + 1.71i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.73891082034967981535128024922, −10.39763042950905848155135076803, −9.497050754564122704043914634338, −8.438754235065555735605438507879, −7.18458080101354326018101529932, −6.26356931623601485222312054185, −5.58209260636933739838989342994, −5.02528969188767107853073572812, −3.37293002575133308890356312071, −2.26044596915791086552431472692,
0.875022971533672723103547542580, 1.96575490099697279108910162980, 2.76268432667033013246204254220, 4.98378928627764820028927656598, 5.70467630231347457442534607253, 6.48536726767020320457339575078, 7.45766221447065655753722253653, 8.141943434734769062899054618719, 9.711924140026074063594863510839, 10.26156919404333790975237579932