| L(s) = 1 | + (−0.939 + 0.342i)2-s + (0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s + (1.57 + 1.31i)5-s + (−0.766 + 0.642i)6-s + (−1.90 + 1.84i)7-s + (−0.500 + 0.866i)8-s + (0.766 − 0.642i)9-s + (−1.92 − 0.702i)10-s + (−0.806 − 1.39i)11-s + (0.500 − 0.866i)12-s + (3.11 − 2.61i)13-s + (1.15 − 2.37i)14-s + (1.92 + 0.702i)15-s + (0.173 − 0.984i)16-s + (3.34 + 2.80i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 + 0.241i)2-s + (0.542 − 0.197i)3-s + (0.383 − 0.321i)4-s + (0.703 + 0.590i)5-s + (−0.312 + 0.262i)6-s + (−0.718 + 0.695i)7-s + (−0.176 + 0.306i)8-s + (0.255 − 0.214i)9-s + (−0.610 − 0.222i)10-s + (−0.243 − 0.421i)11-s + (0.144 − 0.249i)12-s + (0.864 − 0.725i)13-s + (0.309 − 0.636i)14-s + (0.498 + 0.181i)15-s + (0.0434 − 0.246i)16-s + (0.811 + 0.680i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33379 + 0.638553i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33379 + 0.638553i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 7 | \( 1 + (1.90 - 1.84i)T \) |
| 19 | \( 1 + (-2.28 - 3.71i)T \) |
| good | 5 | \( 1 + (-1.57 - 1.31i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (0.806 + 1.39i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.11 + 2.61i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.34 - 2.80i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.537 - 3.04i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.946 - 5.36i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + 1.22T + 31T^{2} \) |
| 37 | \( 1 + (0.205 + 0.356i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.57 - 7.19i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (7.03 - 2.55i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-2.49 + 2.09i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-3.10 + 2.60i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-5.64 - 4.73i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.997 + 5.65i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.58 - 1.67i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (9.51 - 3.46i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (9.68 - 3.52i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-0.0321 + 0.182i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.08 - 1.87i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.99 - 2.54i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-1.87 + 10.6i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.12170773422490115863568196914, −9.616415407969316825765819642603, −8.609181410764620348704552856612, −8.030213539976344570031863169315, −6.97808798243483215818943705529, −5.99143466585747235991441895685, −5.61669077070044461616710189667, −3.50398132465180350862475452320, −2.78937567376060800893339972312, −1.43280816075748998344936586663,
0.975494612438714795943064948378, 2.34060768017445826688043798265, 3.49863542000803435630695241298, 4.58173583482029572904053414082, 5.82776877348754244913684445028, 6.92712223596588358032963509623, 7.62815333390984712119258102840, 8.787613402567656532425996733013, 9.328443377635472325750045865314, 9.935412581875962615569372392977
