| L(s) = 1 | + (1.60 + 1.32i)2-s + (0.307 − 2.43i)3-s + (0.434 + 2.28i)4-s + (3.72 − 3.49i)6-s + (0.457 − 0.629i)7-s + (−0.321 + 0.584i)8-s + (−2.93 − 0.753i)9-s + (2.06 − 2.49i)11-s + (5.68 − 0.357i)12-s + (−0.789 + 3.07i)13-s + (1.56 − 0.402i)14-s + (3.02 − 1.19i)16-s + (−6.10 − 1.16i)17-s + (−3.70 − 5.09i)18-s + (4.38 − 0.553i)19-s + ⋯ |
| L(s) = 1 | + (1.13 + 0.936i)2-s + (0.177 − 1.40i)3-s + (0.217 + 1.14i)4-s + (1.51 − 1.42i)6-s + (0.172 − 0.237i)7-s + (−0.113 + 0.206i)8-s + (−0.977 − 0.251i)9-s + (0.621 − 0.750i)11-s + (1.64 − 0.103i)12-s + (−0.218 + 0.852i)13-s + (0.418 − 0.107i)14-s + (0.756 − 0.299i)16-s + (−1.48 − 0.282i)17-s + (−0.872 − 1.20i)18-s + (1.00 − 0.127i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.86147 - 0.539636i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.86147 - 0.539636i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + (-1.60 - 1.32i)T + (0.374 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-0.307 + 2.43i)T + (-2.90 - 0.746i)T^{2} \) |
| 7 | \( 1 + (-0.457 + 0.629i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-2.06 + 2.49i)T + (-2.06 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.789 - 3.07i)T + (-11.3 - 6.26i)T^{2} \) |
| 17 | \( 1 + (6.10 + 1.16i)T + (15.8 + 6.25i)T^{2} \) |
| 19 | \( 1 + (-4.38 + 0.553i)T + (18.4 - 4.72i)T^{2} \) |
| 23 | \( 1 + (-4.53 + 2.87i)T + (9.79 - 20.8i)T^{2} \) |
| 29 | \( 1 + (2.56 - 5.44i)T + (-18.4 - 22.3i)T^{2} \) |
| 31 | \( 1 + (-0.00238 + 0.0125i)T + (-28.8 - 11.4i)T^{2} \) |
| 37 | \( 1 + (2.70 + 6.84i)T + (-26.9 + 25.3i)T^{2} \) |
| 41 | \( 1 + (6.20 - 9.77i)T + (-17.4 - 37.0i)T^{2} \) |
| 43 | \( 1 + (5.43 - 1.76i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-3.22 - 5.86i)T + (-25.1 + 39.6i)T^{2} \) |
| 53 | \( 1 + (1.64 - 1.75i)T + (-3.32 - 52.8i)T^{2} \) |
| 59 | \( 1 + (0.0986 + 1.56i)T + (-58.5 + 7.39i)T^{2} \) |
| 61 | \( 1 + (-7.16 - 11.2i)T + (-25.9 + 55.1i)T^{2} \) |
| 67 | \( 1 + (8.43 - 3.96i)T + (42.7 - 51.6i)T^{2} \) |
| 71 | \( 1 + (-4.47 + 2.45i)T + (38.0 - 59.9i)T^{2} \) |
| 73 | \( 1 + (-10.0 - 0.634i)T + (72.4 + 9.14i)T^{2} \) |
| 79 | \( 1 + (1.81 + 0.229i)T + (76.5 + 19.6i)T^{2} \) |
| 83 | \( 1 + (-0.488 - 3.86i)T + (-80.3 + 20.6i)T^{2} \) |
| 89 | \( 1 + (0.254 - 4.04i)T + (-88.2 - 11.1i)T^{2} \) |
| 97 | \( 1 + (-1.65 - 0.777i)T + (61.8 + 74.7i)T^{2} \) |
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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
−10.95672926224226505547252574694, −9.335687097943680629660287510989, −8.476024300864384313423606767803, −7.40087044708356796050501123659, −6.85813432764631802423748453301, −6.33533824906435375369802330559, −5.17621961576680273990470995881, −4.17665652385839436635494793020, −2.86891354829741032405165232673, −1.29413576068163312093120876710,
2.01608496672927432310835366927, 3.28001409220531375765159062355, 3.95987366594146078317534825548, 4.92306239212267988382571724606, 5.39488321134213216657487672517, 6.89387407038277115831790751882, 8.312216286195115355112498533824, 9.295888217639412249176243711049, 10.04372925492081557901684482671, 10.73122084249603311392731918513
