L(s) = 1 | + (−1.66 + 0.124i)2-s + (0.137 − 0.444i)3-s + (0.766 − 0.115i)4-s + (−2.19 + 0.436i)5-s + (−0.172 + 0.755i)6-s + (−2.54 + 0.723i)7-s + (1.98 − 0.454i)8-s + (2.30 + 1.56i)9-s + (3.58 − 0.998i)10-s + (4.31 − 2.93i)11-s + (0.0537 − 0.356i)12-s + (1.16 − 2.42i)13-s + (4.13 − 1.51i)14-s + (−0.106 + 1.03i)15-s + (−4.72 + 1.45i)16-s + (1.48 − 0.584i)17-s + ⋯ |
L(s) = 1 | + (−1.17 + 0.0880i)2-s + (0.0791 − 0.256i)3-s + (0.383 − 0.0577i)4-s + (−0.980 + 0.195i)5-s + (−0.0703 + 0.308i)6-s + (−0.961 + 0.273i)7-s + (0.703 − 0.160i)8-s + (0.766 + 0.522i)9-s + (1.13 − 0.315i)10-s + (1.29 − 0.886i)11-s + (0.0155 − 0.102i)12-s + (0.323 − 0.672i)13-s + (1.10 − 0.405i)14-s + (−0.0274 + 0.266i)15-s + (−1.18 + 0.364i)16-s + (0.360 − 0.141i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.547873 - 0.149202i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.547873 - 0.149202i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (2.19 - 0.436i)T \) |
| 7 | \( 1 + (2.54 - 0.723i)T \) |
good | 2 | \( 1 + (1.66 - 0.124i)T + (1.97 - 0.298i)T^{2} \) |
| 3 | \( 1 + (-0.137 + 0.444i)T + (-2.47 - 1.68i)T^{2} \) |
| 11 | \( 1 + (-4.31 + 2.93i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (-1.16 + 2.42i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (-1.48 + 0.584i)T + (12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (2.78 + 4.82i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.85 - 2.69i)T + (16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (-2.50 + 3.13i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (-2.40 + 4.17i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.944 - 6.26i)T + (-35.3 - 10.9i)T^{2} \) |
| 41 | \( 1 + (-1.48 - 6.50i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (3.88 + 0.885i)T + (38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-1.12 + 0.0844i)T + (46.4 - 7.00i)T^{2} \) |
| 53 | \( 1 + (0.703 + 4.66i)T + (-50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (-6.63 - 6.15i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (7.13 + 1.07i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (4.01 + 2.31i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (4.73 + 5.94i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-7.03 - 0.527i)T + (72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + (-1.05 - 1.83i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.43 + 11.2i)T + (-51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (8.42 + 5.74i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + 2.69iT - 97T^{2} \) |
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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
−11.74976309652999746938250920905, −10.92863787012753681488763168854, −9.937394649969204025932108820944, −8.977255304193553948683020911478, −8.245461402485579939061043794171, −7.21414060177148310909599921825, −6.45161040244120823769181518606, −4.50245854588814489635836848570, −3.17059314177594894793602211778, −0.864903002399369166719465541632,
1.23121745796140025705655985347, 3.71835955190658059452867039807, 4.47115016103302395533774775792, 6.72054899133612762759412047077, 7.20978938228421661359629189985, 8.629109080416747039692860304103, 9.217914255950049323066843605004, 10.06345712978716750863060422717, 10.89478910465766275599673809203, 12.20686962475658450002637273420