| L(s) = 1 | + (−2.30 − 14.5i)2-s + (−38.7 − 26.1i)3-s + (−85.1 + 27.6i)4-s + (−102. − 260. i)5-s + (−290. + 625. i)6-s + (−331. + 331. i)7-s + (−257. − 505. i)8-s + (823. + 2.02e3i)9-s + (−3.55e3 + 2.08e3i)10-s + (−1.79e3 + 2.47e3i)11-s + (4.02e3 + 1.15e3i)12-s + (796. + 126. i)13-s + (5.59e3 + 4.06e3i)14-s + (−2.82e3 + 1.27e4i)15-s + (−1.60e4 + 1.16e4i)16-s + (−6.32e3 + 3.22e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.203 − 1.28i)2-s + (−0.829 − 0.558i)3-s + (−0.665 + 0.216i)4-s + (−0.365 − 0.930i)5-s + (−0.549 + 1.18i)6-s + (−0.365 + 0.365i)7-s + (−0.177 − 0.348i)8-s + (0.376 + 0.926i)9-s + (−1.12 + 0.660i)10-s + (−0.406 + 0.560i)11-s + (0.672 + 0.192i)12-s + (0.100 + 0.0159i)13-s + (0.545 + 0.396i)14-s + (−0.216 + 0.976i)15-s + (−0.978 + 0.711i)16-s + (−0.312 + 0.159i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.319724 - 0.0456761i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.319724 - 0.0456761i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (38.7 + 26.1i)T \) |
| 5 | \( 1 + (102. + 260. i)T \) |
| good | 2 | \( 1 + (2.30 + 14.5i)T + (-121. + 39.5i)T^{2} \) |
| 7 | \( 1 + (331. - 331. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (1.79e3 - 2.47e3i)T + (-6.02e6 - 1.85e7i)T^{2} \) |
| 13 | \( 1 + (-796. - 126. i)T + (5.96e7 + 1.93e7i)T^{2} \) |
| 17 | \( 1 + (6.32e3 - 3.22e3i)T + (2.41e8 - 3.31e8i)T^{2} \) |
| 19 | \( 1 + (-3.98e3 - 1.29e3i)T + (7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 + (-2.55e4 + 4.04e3i)T + (3.23e9 - 1.05e9i)T^{2} \) |
| 29 | \( 1 + (3.26e3 + 1.00e4i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-2.31e4 + 7.11e4i)T + (-2.22e10 - 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-8.35e4 + 5.27e5i)T + (-9.02e10 - 2.93e10i)T^{2} \) |
| 41 | \( 1 + (4.39e4 + 6.05e4i)T + (-6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + (4.62e5 + 4.62e5i)T + 2.71e11iT^{2} \) |
| 47 | \( 1 + (5.09e4 - 1.00e5i)T + (-2.97e11 - 4.09e11i)T^{2} \) |
| 53 | \( 1 + (-1.21e6 - 6.17e5i)T + (6.90e11 + 9.50e11i)T^{2} \) |
| 59 | \( 1 + (1.86e6 - 1.35e6i)T + (7.69e11 - 2.36e12i)T^{2} \) |
| 61 | \( 1 + (-2.57e6 - 1.87e6i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 + (-1.64e6 - 3.23e6i)T + (-3.56e12 + 4.90e12i)T^{2} \) |
| 71 | \( 1 + (2.43e6 - 7.90e5i)T + (7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-1.97e5 - 1.24e6i)T + (-1.05e13 + 3.41e12i)T^{2} \) |
| 79 | \( 1 + (1.48e6 - 4.81e5i)T + (1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-2.93e6 - 5.76e6i)T + (-1.59e13 + 2.19e13i)T^{2} \) |
| 89 | \( 1 + (7.85e6 + 5.70e6i)T + (1.36e13 + 4.20e13i)T^{2} \) |
| 97 | \( 1 + (-3.07e6 - 1.56e6i)T + (4.74e13 + 6.53e13i)T^{2} \) |
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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
−12.66195578996012070039814153171, −11.99555177127568945233172558323, −11.07473099281261863724616647686, −9.963435308981124191559007161333, −8.761544589389960421774484068944, −7.21343784021060775475445354178, −5.65189972047641534849369972114, −4.21268755626046510530848421333, −2.30656248073995129918250104828, −1.00888030715751752367797830937,
0.16018806976458090979094023556, 3.26046659366147130104612013043, 4.96384490819316782458102930340, 6.28483717713348920476214298062, 6.95135074565322099586757276999, 8.255145888299093467291734648362, 9.771444871105333213860974881098, 10.90889613655521254766598290601, 11.74884092941327486265961780927, 13.45809331732364097991523331206
