| L(s) = 1 | + (−0.0713 + 0.997i)2-s + (−0.977 + 0.212i)3-s + (−0.989 − 0.142i)4-s + (−2.18 − 0.484i)5-s + (−0.142 − 0.989i)6-s + (−4.31 + 2.35i)7-s + (0.212 − 0.977i)8-s + (0.909 − 0.415i)9-s + (0.638 − 2.14i)10-s + (0.567 + 0.491i)11-s + (0.997 − 0.0713i)12-s + (3.03 − 5.56i)13-s + (−2.04 − 4.47i)14-s + (2.23 + 0.00899i)15-s + (0.959 + 0.281i)16-s + (1.12 + 0.838i)17-s + ⋯ |
| L(s) = 1 | + (−0.0504 + 0.705i)2-s + (−0.564 + 0.122i)3-s + (−0.494 − 0.0711i)4-s + (−0.976 − 0.216i)5-s + (−0.0580 − 0.404i)6-s + (−1.63 + 0.891i)7-s + (0.0751 − 0.345i)8-s + (0.303 − 0.138i)9-s + (0.201 − 0.677i)10-s + (0.171 + 0.148i)11-s + (0.287 − 0.0205i)12-s + (0.842 − 1.54i)13-s + (−0.546 − 1.19i)14-s + (0.577 + 0.00232i)15-s + (0.239 + 0.0704i)16-s + (0.271 + 0.203i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.589302 - 0.0166447i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.589302 - 0.0166447i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0713 - 0.997i)T \) |
| 3 | \( 1 + (0.977 - 0.212i)T \) |
| 5 | \( 1 + (2.18 + 0.484i)T \) |
| 23 | \( 1 + (4.38 + 1.94i)T \) |
| good | 7 | \( 1 + (4.31 - 2.35i)T + (3.78 - 5.88i)T^{2} \) |
| 11 | \( 1 + (-0.567 - 0.491i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.03 + 5.56i)T + (-7.02 - 10.9i)T^{2} \) |
| 17 | \( 1 + (-1.12 - 0.838i)T + (4.78 + 16.3i)T^{2} \) |
| 19 | \( 1 + (0.534 - 3.71i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-6.83 + 0.983i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-6.38 + 4.10i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-3.61 - 1.34i)T + (27.9 + 24.2i)T^{2} \) |
| 41 | \( 1 + (0.469 - 1.02i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.555 - 2.55i)T + (-39.1 + 17.8i)T^{2} \) |
| 47 | \( 1 + (8.63 + 8.63i)T + 47iT^{2} \) |
| 53 | \( 1 + (5.03 + 9.22i)T + (-28.6 + 44.5i)T^{2} \) |
| 59 | \( 1 + (1.73 + 5.89i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (0.0184 + 0.0286i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (-7.55 - 0.540i)T + (66.3 + 9.53i)T^{2} \) |
| 71 | \( 1 + (-2.93 - 3.38i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-5.79 - 7.74i)T + (-20.5 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-11.4 + 3.37i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (0.141 - 0.379i)T + (-62.7 - 54.3i)T^{2} \) |
| 89 | \( 1 + (10.9 + 7.04i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (1.10 + 2.96i)T + (-73.3 + 63.5i)T^{2} \) |
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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
−10.13752156300817417541284064406, −9.765216461241121427799865160513, −8.351068134909380493686272519203, −8.115409244028678289942725135674, −6.61241285846231465273116919751, −6.16183135105588652717713655336, −5.25437053885701415698557124838, −3.95629632156070535919441126685, −3.10315167209918921693686599175, −0.47186334968882854039269285033,
0.947081291067707739881785611727, 2.96118980078132386044065468253, 3.88898743658584887664867490947, 4.57376193016471221811533844497, 6.39267401442235324796985272296, 6.70185919202059699079291708339, 7.83396156627778357954582798224, 9.002048823485211970362154884979, 9.766455070654756939827296245999, 10.64406526547989911610562484129
