| L(s) = 1 | + (1.55 − 0.712i)2-s + (0.478 + 1.63i)3-s + (0.614 − 0.709i)4-s + (1.15 − 1.91i)5-s + (1.90 + 2.20i)6-s + (−4.84 + 0.697i)7-s + (−0.512 + 1.74i)8-s + (0.0940 − 0.0604i)9-s + (0.429 − 3.80i)10-s + (1.53 − 3.35i)11-s + (1.45 + 0.662i)12-s + (−2.46 − 0.354i)13-s + (−7.06 + 4.54i)14-s + (3.67 + 0.958i)15-s + (0.711 + 4.94i)16-s + (2.98 − 2.58i)17-s + ⋯ |
| L(s) = 1 | + (1.10 − 0.503i)2-s + (0.276 + 0.941i)3-s + (0.307 − 0.354i)4-s + (0.514 − 0.857i)5-s + (0.778 + 0.898i)6-s + (−1.83 + 0.263i)7-s + (−0.181 + 0.617i)8-s + (0.0313 − 0.0201i)9-s + (0.135 − 1.20i)10-s + (0.462 − 1.01i)11-s + (0.418 + 0.191i)12-s + (−0.683 − 0.0982i)13-s + (−1.88 + 1.21i)14-s + (0.949 + 0.247i)15-s + (0.177 + 1.23i)16-s + (0.724 − 0.627i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70184 - 0.0437662i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70184 - 0.0437662i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.15 + 1.91i)T \) |
| 23 | \( 1 + (4.46 - 1.74i)T \) |
| good | 2 | \( 1 + (-1.55 + 0.712i)T + (1.30 - 1.51i)T^{2} \) |
| 3 | \( 1 + (-0.478 - 1.63i)T + (-2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (4.84 - 0.697i)T + (6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-1.53 + 3.35i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (2.46 + 0.354i)T + (12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-2.98 + 2.58i)T + (2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (1.98 - 2.29i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-3.40 - 3.92i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-0.607 - 0.178i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (-4.00 - 6.22i)T + (-15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (1.59 + 1.02i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (0.621 + 2.11i)T + (-36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 5.14iT - 47T^{2} \) |
| 53 | \( 1 + (-3.88 + 0.558i)T + (50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.185 - 1.29i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (8.33 + 2.44i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-12.2 + 5.59i)T + (43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-0.908 - 1.98i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (3.52 + 3.05i)T + (10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (1.02 - 7.13i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (5.15 + 8.01i)T + (-34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (5.44 - 1.60i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-2.18 + 3.40i)T + (-40.2 - 88.2i)T^{2} \) |
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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
−13.49451949575134171567422218025, −12.56833143711923990806733324212, −11.97786378021402125106602954251, −10.22855024203929992025988777163, −9.546558351724792356815291464179, −8.589582321125732761951223553232, −6.30664324618341176565901497068, −5.27370971666401521965717142913, −3.93364750673078558903755246055, −3.01092513209604098531828702801,
2.63518385904627759195710803484, 4.11415596932878539167497129053, 6.07065685132661859692859340463, 6.68859368389884520428974486625, 7.42234795027865696196599150537, 9.637821519430633162178777847328, 10.14796608282554664618842067253, 12.29278496004590584619755511145, 12.77111929480484408516942071412, 13.58525248656636578254095499390
