| L(s) = 1 | + (0.112 + 2.15i)2-s + (1.03 + 0.839i)3-s + (−2.63 + 0.277i)4-s + (−1.72 + 1.42i)5-s + (−1.69 + 2.32i)6-s + (−2.35 − 1.19i)7-s + (−0.220 − 1.39i)8-s + (−0.253 − 1.19i)9-s + (−3.26 − 3.54i)10-s + (4.15 + 0.882i)11-s + (−2.96 − 1.92i)12-s + (2.42 + 4.75i)13-s + (2.31 − 5.21i)14-s + (−2.98 + 0.0321i)15-s + (−2.21 + 0.470i)16-s + (4.69 + 1.80i)17-s + ⋯ |
| L(s) = 1 | + (0.0798 + 1.52i)2-s + (0.598 + 0.484i)3-s + (−1.31 + 0.138i)4-s + (−0.770 + 0.637i)5-s + (−0.690 + 0.950i)6-s + (−0.891 − 0.453i)7-s + (−0.0780 − 0.492i)8-s + (−0.0846 − 0.398i)9-s + (−1.03 − 1.12i)10-s + (1.25 + 0.266i)11-s + (−0.856 − 0.556i)12-s + (0.671 + 1.31i)13-s + (0.619 − 1.39i)14-s + (−0.769 + 0.00828i)15-s + (−0.553 + 0.117i)16-s + (1.13 + 0.437i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.256i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.154266 + 1.18419i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.154266 + 1.18419i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.72 - 1.42i)T \) |
| 7 | \( 1 + (2.35 + 1.19i)T \) |
| good | 2 | \( 1 + (-0.112 - 2.15i)T + (-1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (-1.03 - 0.839i)T + (0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-4.15 - 0.882i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-2.42 - 4.75i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-4.69 - 1.80i)T + (12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.174 + 1.66i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.59 + 0.0835i)T + (22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (4.43 + 6.10i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.987 - 2.21i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (1.89 - 2.92i)T + (-15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-3.63 - 1.18i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (4.20 + 4.20i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.17 - 8.27i)T + (-34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (-2.88 + 3.56i)T + (-11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (4.69 + 5.21i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (8.51 + 7.66i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-3.28 + 8.56i)T + (-49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-11.8 + 8.63i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.58 - 1.67i)T + (29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (3.71 - 8.33i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (0.313 - 0.0496i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (3.99 - 4.43i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (4.27 + 0.676i)T + (92.2 + 29.9i)T^{2} \) |
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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
−13.86421051092961283516828082522, −12.27057949564615520180106853865, −11.20283808733927587071333961305, −9.675151201175400591700848583801, −8.997551027513211052006333306087, −7.86259813115828581830674198368, −6.74819445781092836773698271701, −6.30062453637559821722558751408, −4.25959772982842677474446230618, −3.53198554685277714727366189661,
1.18474261205631785691260234782, 3.01926342323173275096561636690, 3.75614779281628543462862715626, 5.51432523123947159618155110587, 7.30543176695276189734744320771, 8.550096536746381053188776873604, 9.255816237574337720839495955427, 10.42848474358320151605014335420, 11.49026864187711518237651145292, 12.33514161313440747300450100796
