L(s) = 1 | + (3 − 5.19i)2-s + (4.5 + 7.79i)3-s + (−2 − 3.46i)4-s + (−39 + 67.5i)5-s + 54·6-s + 168·8-s + (−40.5 + 70.1i)9-s + (234 + 405. i)10-s + (−222 − 384. i)11-s + (18.0 − 31.1i)12-s − 442·13-s − 702·15-s + (568 − 983. i)16-s + (63 + 109. i)17-s + (243 + 420. i)18-s + (−1.34e3 + 2.32e3i)19-s + ⋯ |
L(s) = 1 | + (0.530 − 0.918i)2-s + (0.288 + 0.499i)3-s + (−0.0625 − 0.108i)4-s + (−0.697 + 1.20i)5-s + 0.612·6-s + 0.928·8-s + (−0.166 + 0.288i)9-s + (0.739 + 1.28i)10-s + (−0.553 − 0.958i)11-s + (0.0360 − 0.0625i)12-s − 0.725·13-s − 0.805·15-s + (0.554 − 0.960i)16-s + (0.0528 + 0.0915i)17-s + (0.176 + 0.306i)18-s + (−0.852 + 1.47i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.461101261\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.461101261\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-4.5 - 7.79i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-3 + 5.19i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (39 - 67.5i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (222 + 384. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 442T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-63 - 109. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (1.34e3 - 2.32e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (2.10e3 - 3.63e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 5.44e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (40 + 69.2i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-2.71e3 + 4.70e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 7.96e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.15e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-6.96e3 + 1.20e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-4.79e3 - 8.30e3i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.37e4 + 2.38e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.47e4 - 4.28e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.96e4 - 5.14e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 3.20e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-3.09e4 - 5.35e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-3.28e4 + 5.69e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 4.01e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-3.98e3 + 6.90e3i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.43e5T + 8.58e9T^{2} \) |
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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
−12.25735097790308652579408349638, −11.33722347091717048290249089610, −10.70787242846637690066120869274, −9.866477786776292022879285271579, −8.110807781561089577122431050035, −7.37058665093314473129618239347, −5.67061899817867306119845861452, −3.97110970669868766338939510135, −3.35684217607051297998196800219, −2.18573393359572254728069463825,
0.37103823368060179196664231475, 2.12192349625375215475327525950, 4.39714423639403199621530903871, 5.00084021470008397935389571515, 6.50131624677641011352285160093, 7.52306568955418276409634218132, 8.268769414663246479157885117185, 9.504787738818300343770048651040, 10.92813965320586952857455138305, 12.37267942221707701866710971737