| L(s) = 1 | + (−0.379 + 1.66i)2-s + (−2.24 + 2.81i)3-s + (4.58 + 2.20i)4-s + (−4.66 + 5.84i)5-s + (−3.82 − 4.79i)6-s + (−14.0 − 12.0i)7-s + (−13.9 + 17.4i)8-s + (3.12 + 13.7i)9-s + (−7.95 − 9.97i)10-s + (−7.78 + 34.1i)11-s + (−16.4 + 7.94i)12-s + (5.32 − 23.3i)13-s + (25.4 − 18.7i)14-s + (−5.98 − 26.2i)15-s + (1.63 + 2.05i)16-s + (75.9 − 36.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.134 + 0.587i)2-s + (−0.431 + 0.541i)3-s + (0.573 + 0.276i)4-s + (−0.417 + 0.523i)5-s + (−0.260 − 0.326i)6-s + (−0.758 − 0.652i)7-s + (−0.615 + 0.771i)8-s + (0.115 + 0.507i)9-s + (−0.251 − 0.315i)10-s + (−0.213 + 0.934i)11-s + (−0.396 + 0.191i)12-s + (0.113 − 0.497i)13-s + (0.485 − 0.358i)14-s + (−0.103 − 0.451i)15-s + (0.0256 + 0.0321i)16-s + (1.08 − 0.522i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.655i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.754 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.363025 + 0.971189i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.363025 + 0.971189i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (14.0 + 12.0i)T \) |
| good | 2 | \( 1 + (0.379 - 1.66i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (2.24 - 2.81i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (4.66 - 5.84i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (7.78 - 34.1i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-5.32 + 23.3i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (-75.9 + 36.5i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 - 57.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-139. - 67.1i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (-98.6 + 47.5i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 + 60.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-178. + 85.8i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (233. - 293. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (307. + 385. i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (-14.5 + 63.8i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (-384. - 185. i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (220. + 276. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (-542. + 261. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 - 49.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + (40.7 + 19.6i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (168. + 738. i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 + 513.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (22.1 + 97.1i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-253. - 1.10e3i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 - 853.T + 9.12e5T^{2} \) |
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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
−15.68307631971271329225897790751, −14.89180040601797272884495823246, −13.30545486396323940140775013732, −11.89105461063397787743837217592, −10.79348107077638191838164463712, −9.774398239016699985726130641296, −7.70713836000775726300230745784, −6.98940534109578668101754254703, −5.29091233646664862728516106556, −3.25268668888693359787850582440,
0.909220272610984490028241901184, 3.20826064564170579168082391051, 5.76594711278810446970765258519, 6.84971775311214898280667951335, 8.701369079882567843125816062332, 10.00572197816471138534905664968, 11.45857923534011006746288628818, 12.18359598450924833457551409573, 13.00369830840227411452624744196, 14.80653989433147448144774385322
