| L(s) = 1 | + (−6.01 − 4.37i)2-s + (1.80 + 5.55i)3-s + (7.21 + 22.2i)4-s + (−36.8 + 42.0i)5-s + (13.4 − 41.3i)6-s + 204.·7-s + (−19.8 + 61.1i)8-s + (169. − 122. i)9-s + (405. − 91.7i)10-s + (383. + 278. i)11-s + (−110. + 80.1i)12-s + (−542. + 393. i)13-s + (−1.23e3 − 894. i)14-s + (−299. − 128. i)15-s + (991. − 720. i)16-s + (−605. + 1.86e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.06 − 0.773i)2-s + (0.115 + 0.356i)3-s + (0.225 + 0.694i)4-s + (−0.659 + 0.751i)5-s + (0.152 − 0.468i)6-s + 1.57·7-s + (−0.109 + 0.338i)8-s + (0.695 − 0.505i)9-s + (1.28 − 0.290i)10-s + (0.956 + 0.694i)11-s + (−0.221 + 0.160i)12-s + (−0.889 + 0.646i)13-s + (−1.67 − 1.21i)14-s + (−0.344 − 0.147i)15-s + (0.968 − 0.703i)16-s + (−0.508 + 1.56i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.884962 + 0.142526i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.884962 + 0.142526i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (36.8 - 42.0i)T \) |
| good | 2 | \( 1 + (6.01 + 4.37i)T + (9.88 + 30.4i)T^{2} \) |
| 3 | \( 1 + (-1.80 - 5.55i)T + (-196. + 142. i)T^{2} \) |
| 7 | \( 1 - 204.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-383. - 278. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (542. - 393. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (605. - 1.86e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-95.9 + 295. i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-1.74e3 - 1.26e3i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (840. + 2.58e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (115. - 354. i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (1.01e3 - 737. i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (73.7 - 53.5i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 1.21e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (8.31e3 + 2.56e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (2.26e3 + 6.95e3i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (2.03e4 - 1.47e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-2.84e4 - 2.06e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (1.15e4 - 3.54e4i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-2.79e3 - 8.61e3i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (1.40e4 + 1.02e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (1.64e4 + 5.05e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-4.82e3 + 1.48e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (8.41e4 + 6.11e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-1.28e4 - 3.94e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−17.20010731496314640449965700560, −15.02661972786604028841766920120, −14.64549527606694549800842826372, −12.05003905310306820197614797789, −11.20384449931499953569400609692, −10.05625577412412557304058404387, −8.704281314126198913061835570931, −7.26166634167760727795760606782, −4.27054159048042931102870642828, −1.74712187802416702906884025984,
0.987746712766938554763095806759, 4.80032718409981152154733662628, 7.27024049913664093355690967946, 8.087408374540984416089225697216, 9.212220614297008813902554784040, 11.16254064125412127132651292246, 12.54539298752162306043002631976, 14.26821020300937148614194234775, 15.60329279628465920224571690789, 16.65248440945926871578988489059
