L(s) = 1 | + (4.57 + 6.30i)2-s + (3.07 + 0.997i)3-s + (−8.86 + 27.2i)4-s + (10.4 + 54.9i)5-s + (7.77 + 23.9i)6-s + 46.1i·7-s + (24.4 − 7.93i)8-s + (−188. − 136. i)9-s + (−298. + 317. i)10-s + (144. − 104. i)11-s + (−54.4 + 74.9i)12-s + (267. − 368. i)13-s + (−290. + 211. i)14-s + (−22.5 + 179. i)15-s + (904. + 657. i)16-s + (1.37e3 − 445. i)17-s + ⋯ |
L(s) = 1 | + (0.809 + 1.11i)2-s + (0.196 + 0.0640i)3-s + (−0.277 + 0.853i)4-s + (0.187 + 0.982i)5-s + (0.0881 + 0.271i)6-s + 0.355i·7-s + (0.134 − 0.0438i)8-s + (−0.774 − 0.562i)9-s + (−0.942 + 1.00i)10-s + (0.358 − 0.260i)11-s + (−0.109 + 0.150i)12-s + (0.439 − 0.605i)13-s + (−0.396 + 0.288i)14-s + (−0.0259 + 0.205i)15-s + (0.883 + 0.642i)16-s + (1.15 − 0.374i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.47523 + 1.78367i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.47523 + 1.78367i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-10.4 - 54.9i)T \) |
good | 2 | \( 1 + (-4.57 - 6.30i)T + (-9.88 + 30.4i)T^{2} \) |
| 3 | \( 1 + (-3.07 - 0.997i)T + (196. + 142. i)T^{2} \) |
| 7 | \( 1 - 46.1iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-144. + 104. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-267. + 368. i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-1.37e3 + 445. i)T + (1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (68.7 + 211. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (2.09e3 + 2.88e3i)T + (-1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (269. - 830. i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-520. - 1.60e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (8.11e3 - 1.11e4i)T + (-2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (8.74e3 + 6.35e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 2.74e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (2.11e4 + 6.88e3i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (6.80e3 + 2.21e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-3.72e4 - 2.70e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.60e4 - 1.16e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-4.75e4 + 1.54e4i)T + (1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (1.32e4 - 4.09e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (2.83e4 + 3.90e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-2.72e4 + 8.37e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-6.30e4 + 2.04e4i)T + (3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (8.13e4 - 5.90e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (1.98e4 + 6.43e3i)T + (6.94e9 + 5.04e9i)T^{2} \) |
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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
−16.57462333511242551033736029418, −15.29163003800170788137334956490, −14.52609561155957846571995601408, −13.70795688193726084861820993601, −11.98103002041820090721824193480, −10.28119870054273635177895742015, −8.315205002008278030124962512110, −6.69527638319911027584210234639, −5.62405521237971451171633024990, −3.36227108266422030002658160991,
1.68160506342705082412867025970, 3.82143841961905378969539658673, 5.41591001495219990642070864112, 8.069121877071693991149859759028, 9.769864750020051334132385777690, 11.31858402086396710044249847728, 12.34415992798999906549653080713, 13.51910683103467312133700099974, 14.29140593580586247876907759772, 16.33377015443421664870048098063