| L(s) = 1 | + (−39.0 − 10.4i)2-s + (−2.23 − 8.35i)3-s + (968. + 559. i)4-s + (42.9 − 1.39e3i)5-s + 349. i·6-s + (6.35e3 − 99.7i)7-s + (−1.73e4 − 1.73e4i)8-s + (1.69e4 − 9.80e3i)9-s + (−1.62e4 + 5.40e4i)10-s + (−3.65e4 + 6.33e4i)11-s + (2.50e3 − 9.34e3i)12-s + (−8.32e4 + 8.32e4i)13-s + (−2.48e5 − 6.24e4i)14-s + (−1.17e4 + 2.76e3i)15-s + (2.08e5 + 3.60e5i)16-s + (2.04e5 − 5.46e4i)17-s + ⋯ |
| L(s) = 1 | + (−1.72 − 0.461i)2-s + (−0.0159 − 0.0595i)3-s + (1.89 + 1.09i)4-s + (0.0307 − 0.999i)5-s + 0.110i·6-s + (0.999 − 0.0157i)7-s + (−1.49 − 1.49i)8-s + (0.862 − 0.498i)9-s + (−0.514 + 1.70i)10-s + (−0.753 + 1.30i)11-s + (0.0348 − 0.130i)12-s + (−0.808 + 0.808i)13-s + (−1.73 − 0.434i)14-s + (−0.0600 + 0.0141i)15-s + (0.794 + 1.37i)16-s + (0.592 − 0.158i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.427i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.903 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.888966 - 0.199657i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.888966 - 0.199657i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-42.9 + 1.39e3i)T \) |
| 7 | \( 1 + (-6.35e3 + 99.7i)T \) |
| good | 2 | \( 1 + (39.0 + 10.4i)T + (443. + 256i)T^{2} \) |
| 3 | \( 1 + (2.23 + 8.35i)T + (-1.70e4 + 9.84e3i)T^{2} \) |
| 11 | \( 1 + (3.65e4 - 6.33e4i)T + (-1.17e9 - 2.04e9i)T^{2} \) |
| 13 | \( 1 + (8.32e4 - 8.32e4i)T - 1.06e10iT^{2} \) |
| 17 | \( 1 + (-2.04e5 + 5.46e4i)T + (1.02e11 - 5.92e10i)T^{2} \) |
| 19 | \( 1 + (-4.70e5 - 8.14e5i)T + (-1.61e11 + 2.79e11i)T^{2} \) |
| 23 | \( 1 + (-5.14e5 + 1.91e6i)T + (-1.55e12 - 9.00e11i)T^{2} \) |
| 29 | \( 1 - 3.47e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + (-1.49e6 - 8.64e5i)T + (1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-1.43e7 - 3.84e6i)T + (1.12e14 + 6.49e13i)T^{2} \) |
| 41 | \( 1 - 1.48e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + (-1.16e7 - 1.16e7i)T + 5.02e14iT^{2} \) |
| 47 | \( 1 + (-1.43e7 + 5.36e7i)T + (-9.69e14 - 5.59e14i)T^{2} \) |
| 53 | \( 1 + (-2.53e6 + 6.79e5i)T + (2.85e15 - 1.64e15i)T^{2} \) |
| 59 | \( 1 + (-2.62e7 + 4.54e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (2.12e7 - 1.22e7i)T + (5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-1.15e7 - 4.32e7i)T + (-2.35e16 + 1.36e16i)T^{2} \) |
| 71 | \( 1 - 1.14e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (6.20e6 + 2.31e7i)T + (-5.09e16 + 2.94e16i)T^{2} \) |
| 79 | \( 1 + (-1.54e8 + 8.89e7i)T + (5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-1.56e8 + 1.56e8i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 + (1.36e8 + 2.35e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (3.93e8 + 3.93e8i)T + 7.60e17iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.74820801600234224131435318615, −12.52799191305015238052290030038, −11.90509778880590793159259906065, −10.22872305153090421913824789229, −9.490182804608087771951441033949, −8.140994225313820467013568462293, −7.24664767473874015041007231865, −4.69467890257363859738127053103, −1.99727736082270085274967553610, −1.00940024881734890118198504247,
0.812894806976958046676950981351, 2.55467755798179232815678328169, 5.58384166032363074569927315125, 7.41769777269863318763545372741, 7.86394743929467218023027159505, 9.561241990747825124609549323207, 10.66887606997572513979636768133, 11.32017961382576745495379669788, 13.69273870961245565144247270517, 15.19788526713943095273062666783
