| L(s) = 1 | + (1.83 − 1.69i)2-s + (−7.79 + 1.17i)3-s + (−0.131 + 1.75i)4-s + (−6.62 + 16.8i)5-s + (−12.2 + 15.3i)6-s + (−4.66 − 17.9i)7-s + (15.1 + 19.0i)8-s + (33.5 − 10.3i)9-s + (16.5 + 42.1i)10-s + (−42.9 − 13.2i)11-s + (−1.03 − 13.7i)12-s + (8.81 + 38.6i)13-s + (−39.0 − 24.9i)14-s + (31.7 − 139. i)15-s + (46.3 + 6.98i)16-s + (−37.1 + 25.3i)17-s + ⋯ |
| L(s) = 1 | + (0.647 − 0.600i)2-s + (−1.49 + 0.226i)3-s + (−0.0164 + 0.218i)4-s + (−0.592 + 1.51i)5-s + (−0.835 + 1.04i)6-s + (−0.251 − 0.967i)7-s + (0.671 + 0.842i)8-s + (1.24 − 0.383i)9-s + (0.523 + 1.33i)10-s + (−1.17 − 0.362i)11-s + (−0.0248 − 0.331i)12-s + (0.188 + 0.824i)13-s + (−0.744 − 0.475i)14-s + (0.547 − 2.39i)15-s + (0.724 + 0.109i)16-s + (−0.530 + 0.361i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.410209 + 0.572434i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.410209 + 0.572434i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (4.66 + 17.9i)T \) |
| good | 2 | \( 1 + (-1.83 + 1.69i)T + (0.597 - 7.97i)T^{2} \) |
| 3 | \( 1 + (7.79 - 1.17i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (6.62 - 16.8i)T + (-91.6 - 85.0i)T^{2} \) |
| 11 | \( 1 + (42.9 + 13.2i)T + (1.09e3 + 749. i)T^{2} \) |
| 13 | \( 1 + (-8.81 - 38.6i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (37.1 - 25.3i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (-8.67 - 15.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-131. - 89.4i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (-134. - 64.8i)T + (1.52e4 + 1.90e4i)T^{2} \) |
| 31 | \( 1 + (-43.8 + 75.8i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (4.16 + 55.6i)T + (-5.00e4 + 7.54e3i)T^{2} \) |
| 41 | \( 1 + (105. + 132. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (274. - 344. i)T + (-1.76e4 - 7.75e4i)T^{2} \) |
| 47 | \( 1 + (238. - 221. i)T + (7.75e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (46.5 - 621. i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (-123. - 313. i)T + (-1.50e5 + 1.39e5i)T^{2} \) |
| 61 | \( 1 + (24.3 + 325. i)T + (-2.24e5 + 3.38e4i)T^{2} \) |
| 67 | \( 1 + (-266. + 461. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-737. + 354. i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (-24.7 - 22.9i)T + (2.90e4 + 3.87e5i)T^{2} \) |
| 79 | \( 1 + (386. + 669. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-43.1 + 189. i)T + (-5.15e5 - 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-847. + 261. i)T + (5.82e5 - 3.97e5i)T^{2} \) |
| 97 | \( 1 - 365.T + 9.12e5T^{2} \) |
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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
−15.56520316087524969548820303412, −14.06242380844186670182892607912, −12.99429929795355848801772136645, −11.62005719919380609314703317941, −10.99371664361915453927959531286, −10.42780534475270936509713422510, −7.60053278711977054176028094494, −6.47742584730708322835522703062, −4.68595053014469955984807641585, −3.27916148642208675825817118949,
0.52476982214939754760309454839, 5.03868084360998650983118463238, 5.17526133827957166133644751600, 6.67846967375426322509072887092, 8.416260501707420015184089828295, 10.17216496011618661193297580270, 11.60214891085904076679080721860, 12.71803404600770685151492609175, 13.08536617121201120831814617636, 15.31515272679041273853463028840
