| L(s) = 1 | + (1.87 + 0.684i)2-s + (3.83 − 3.50i)3-s + (3.06 + 2.57i)4-s + (−3.01 + 17.0i)5-s + (9.60 − 3.97i)6-s + (16.7 − 14.0i)7-s + (4.00 + 6.92i)8-s + (2.37 − 26.8i)9-s + (−17.3 + 30.0i)10-s + (−7.05 − 39.9i)11-s + (20.7 − 0.898i)12-s + (−46.7 + 17.0i)13-s + (40.9 − 14.9i)14-s + (48.3 + 76.0i)15-s + (2.77 + 15.7i)16-s + (−64.5 + 111. i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.737 − 0.675i)3-s + (0.383 + 0.321i)4-s + (−0.269 + 1.52i)5-s + (0.653 − 0.270i)6-s + (0.901 − 0.756i)7-s + (0.176 + 0.306i)8-s + (0.0879 − 0.996i)9-s + (−0.548 + 0.949i)10-s + (−0.193 − 1.09i)11-s + (0.499 − 0.0216i)12-s + (−0.997 + 0.362i)13-s + (0.782 − 0.284i)14-s + (0.832 + 1.30i)15-s + (0.0434 + 0.246i)16-s + (−0.920 + 1.59i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.32857 + 0.219359i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.32857 + 0.219359i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.87 - 0.684i)T \) |
| 3 | \( 1 + (-3.83 + 3.50i)T \) |
| good | 5 | \( 1 + (3.01 - 17.0i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (-16.7 + 14.0i)T + (59.5 - 337. i)T^{2} \) |
| 11 | \( 1 + (7.05 + 39.9i)T + (-1.25e3 + 455. i)T^{2} \) |
| 13 | \( 1 + (46.7 - 17.0i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (64.5 - 111. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (36.0 + 62.3i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (14.7 + 12.3i)T + (2.11e3 + 1.19e4i)T^{2} \) |
| 29 | \( 1 + (-186. - 67.8i)T + (1.86e4 + 1.56e4i)T^{2} \) |
| 31 | \( 1 + (37.3 + 31.3i)T + (5.17e3 + 2.93e4i)T^{2} \) |
| 37 | \( 1 + (-137. + 238. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-6.64 + 2.41i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-14.1 - 80.3i)T + (-7.47e4 + 2.71e4i)T^{2} \) |
| 47 | \( 1 + (11.4 - 9.57i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 - 354.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-26.8 + 152. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (170. - 143. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-458. + 167. i)T + (2.30e5 - 1.93e5i)T^{2} \) |
| 71 | \( 1 + (503. - 872. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-465. - 806. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-302. - 110. i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (56.1 + 20.4i)T + (4.38e5 + 3.67e5i)T^{2} \) |
| 89 | \( 1 + (171. + 296. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-61.1 - 346. i)T + (-8.57e5 + 3.12e5i)T^{2} \) |
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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
−14.52889679189461795547054830382, −14.11494744425416445053683221680, −12.94571353641044098585726153025, −11.41864981023721410219426831970, −10.59716564201620624473715462638, −8.401829004811644409307594396401, −7.32423037524469345645767837295, −6.41498844419001921329113062557, −4.01974473391881157103672979401, −2.50551074360838790373127656934,
2.25227448283208566595722801626, 4.63164689806860980370844077589, 4.99683278596597665891106891932, 7.75198151922699636641294082594, 8.889788013341283746113149161767, 9.976617669006184641055970276541, 11.72113575804046908904361234183, 12.51983345579649007793133419371, 13.71442630424233945166451545324, 14.94106771825300525603798895841
