| L(s) = 1 | + (1.86 − 0.732i)2-s + (5.06 + 1.16i)3-s + (−2.91 + 2.70i)4-s + (7.00 − 4.77i)5-s + (10.3 − 1.52i)6-s + (10.3 − 15.3i)7-s + (−10.4 + 21.6i)8-s + (24.2 + 11.8i)9-s + (9.56 − 14.0i)10-s + (4.91 − 32.5i)11-s + (−17.9 + 10.3i)12-s + (46.1 + 36.8i)13-s + (8.02 − 36.2i)14-s + (41.0 − 15.9i)15-s + (−1.21 + 16.2i)16-s + (−46.7 + 14.4i)17-s + ⋯ |
| L(s) = 1 | + (0.659 − 0.258i)2-s + (0.974 + 0.224i)3-s + (−0.364 + 0.338i)4-s + (0.626 − 0.426i)5-s + (0.700 − 0.103i)6-s + (0.557 − 0.829i)7-s + (−0.460 + 0.956i)8-s + (0.898 + 0.438i)9-s + (0.302 − 0.443i)10-s + (0.134 − 0.893i)11-s + (−0.431 + 0.247i)12-s + (0.985 + 0.786i)13-s + (0.153 − 0.691i)14-s + (0.706 − 0.275i)15-s + (−0.0189 + 0.253i)16-s + (−0.667 + 0.205i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.26979 - 0.192598i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.26979 - 0.192598i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-5.06 - 1.16i)T \) |
| 7 | \( 1 + (-10.3 + 15.3i)T \) |
| good | 2 | \( 1 + (-1.86 + 0.732i)T + (5.86 - 5.44i)T^{2} \) |
| 5 | \( 1 + (-7.00 + 4.77i)T + (45.6 - 116. i)T^{2} \) |
| 11 | \( 1 + (-4.91 + 32.5i)T + (-1.27e3 - 392. i)T^{2} \) |
| 13 | \( 1 + (-46.1 - 36.8i)T + (488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + (46.7 - 14.4i)T + (4.05e3 - 2.76e3i)T^{2} \) |
| 19 | \( 1 + (-57.6 - 33.2i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-7.82 + 25.3i)T + (-1.00e4 - 6.85e3i)T^{2} \) |
| 29 | \( 1 + (260. + 59.3i)T + (2.19e4 + 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-39.4 + 22.7i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (182. + 169. i)T + (3.78e3 + 5.05e4i)T^{2} \) |
| 41 | \( 1 + (367. + 176. i)T + (4.29e4 + 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-82.4 + 39.6i)T + (4.95e4 - 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-32.5 - 82.8i)T + (-7.61e4 + 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-504. - 543. i)T + (-1.11e4 + 1.48e5i)T^{2} \) |
| 59 | \( 1 + (317. + 216. i)T + (7.50e4 + 1.91e5i)T^{2} \) |
| 61 | \( 1 + (342. - 369. i)T + (-1.69e4 - 2.26e5i)T^{2} \) |
| 67 | \( 1 + (40.5 + 70.1i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (815. - 186. i)T + (3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (834. + 327. i)T + (2.85e5 + 2.64e5i)T^{2} \) |
| 79 | \( 1 + (257. - 445. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-417. - 523. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-939. + 141. i)T + (6.73e5 - 2.07e5i)T^{2} \) |
| 97 | \( 1 - 206. iT - 9.12e5T^{2} \) |
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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
−13.05303473275456570773764479598, −11.63846057057776652576605891240, −10.62302939892369494369338952262, −9.162213774823605407840308692383, −8.643489416669630605402718307659, −7.44997796087760794354887124917, −5.66215450294569434950242612637, −4.30513610310144869918902681737, −3.50842567204696140453299698719, −1.72756427838838284244646797051,
1.75456870302303485726199937831, 3.28727226226698787865454834945, 4.79435213889140197993072319536, 5.96986339586023850911385670071, 7.13048789125637129674022086411, 8.556326564496611361421662675282, 9.400032495335716697268279548768, 10.33972412577692041549062273366, 11.89027226700285322413760713955, 13.12828911747958479098956081946
