| L(s) = 1 | + (−4.97 − 1.95i)2-s + (3.58 + 3.76i)3-s + (15.0 + 13.9i)4-s + (2.11 + 1.44i)5-s + (−10.4 − 25.7i)6-s + (−9.97 − 15.6i)7-s + (−29.1 − 60.4i)8-s + (−1.33 + 26.9i)9-s + (−7.70 − 11.3i)10-s + (−8.12 − 53.8i)11-s + (1.35 + 106. i)12-s + (15.0 − 12.0i)13-s + (19.1 + 97.0i)14-s + (2.14 + 13.1i)15-s + (14.5 + 193. i)16-s + (110. + 34.1i)17-s + ⋯ |
| L(s) = 1 | + (−1.75 − 0.690i)2-s + (0.689 + 0.724i)3-s + (1.88 + 1.74i)4-s + (0.189 + 0.129i)5-s + (−0.712 − 1.74i)6-s + (−0.538 − 0.842i)7-s + (−1.28 − 2.67i)8-s + (−0.0494 + 0.998i)9-s + (−0.243 − 0.357i)10-s + (−0.222 − 1.47i)11-s + (0.0325 + 2.56i)12-s + (0.322 − 0.256i)13-s + (0.365 + 1.85i)14-s + (0.0370 + 0.226i)15-s + (0.226 + 3.02i)16-s + (1.57 + 0.486i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.569 + 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.769737 - 0.403399i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.769737 - 0.403399i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.58 - 3.76i)T \) |
| 7 | \( 1 + (9.97 + 15.6i)T \) |
| good | 2 | \( 1 + (4.97 + 1.95i)T + (5.86 + 5.44i)T^{2} \) |
| 5 | \( 1 + (-2.11 - 1.44i)T + (45.6 + 116. i)T^{2} \) |
| 11 | \( 1 + (8.12 + 53.8i)T + (-1.27e3 + 392. i)T^{2} \) |
| 13 | \( 1 + (-15.0 + 12.0i)T + (488. - 2.14e3i)T^{2} \) |
| 17 | \( 1 + (-110. - 34.1i)T + (4.05e3 + 2.76e3i)T^{2} \) |
| 19 | \( 1 + (11.5 - 6.65i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (20.5 + 66.6i)T + (-1.00e4 + 6.85e3i)T^{2} \) |
| 29 | \( 1 + (-215. + 49.0i)T + (2.19e4 - 1.05e4i)T^{2} \) |
| 31 | \( 1 + (-96.7 - 55.8i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-192. + 178. i)T + (3.78e3 - 5.05e4i)T^{2} \) |
| 41 | \( 1 + (54.3 - 26.1i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-136. - 65.5i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (14.6 - 37.2i)T + (-7.61e4 - 7.06e4i)T^{2} \) |
| 53 | \( 1 + (-426. + 459. i)T + (-1.11e4 - 1.48e5i)T^{2} \) |
| 59 | \( 1 + (297. - 203. i)T + (7.50e4 - 1.91e5i)T^{2} \) |
| 61 | \( 1 + (-50.1 - 54.0i)T + (-1.69e4 + 2.26e5i)T^{2} \) |
| 67 | \( 1 + (-301. + 522. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (510. + 116. i)T + (3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (-907. + 356. i)T + (2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (359. + 622. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-407. + 510. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (351. + 53.0i)T + (6.73e5 + 2.07e5i)T^{2} \) |
| 97 | \( 1 - 1.13e3iT - 9.12e5T^{2} \) |
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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
−12.03738149734452049943960670740, −10.64953865972488091283326832162, −10.42477689821818577814381192163, −9.499436740589629312500934133490, −8.345661273382403105359917328574, −7.86952729881547918099911596860, −6.26431696592769887802295948471, −3.69184881524268180211852594057, −2.76865467899111994679889177976, −0.78458702418158078779545931412,
1.30371740997668117956797441099, 2.60999204776177382099857304865, 5.66481905819521701908832649092, 6.78707693377714018165449837949, 7.61615857577198876082544913369, 8.534129415827349635518490344838, 9.579289433514368792579861066846, 9.922672393821647898730146008766, 11.70074706039052944299042484389, 12.53426616115268967482512045201
