| L(s) = 1 | + (−3.13 + 2.27i)2-s + (2.57 + 7.91i)3-s + (2.16 − 6.64i)4-s + (−14.0 − 10.1i)5-s + (−26.0 − 18.9i)6-s + (−2.16 + 6.65i)7-s + (−1.20 − 3.71i)8-s + (−34.1 + 24.8i)9-s + 67.0·10-s + (35.0 − 10.2i)11-s + 58.1·12-s + (−45.3 + 32.9i)13-s + (−8.37 − 25.7i)14-s + (44.5 − 137. i)15-s + (57.4 + 41.7i)16-s + (−35.4 − 25.7i)17-s + ⋯ |
| L(s) = 1 | + (−1.10 + 0.804i)2-s + (0.494 + 1.52i)3-s + (0.270 − 0.831i)4-s + (−1.25 − 0.910i)5-s + (−1.77 − 1.28i)6-s + (−0.116 + 0.359i)7-s + (−0.0533 − 0.164i)8-s + (−1.26 + 0.919i)9-s + 2.12·10-s + (0.959 − 0.280i)11-s + 1.39·12-s + (−0.967 + 0.702i)13-s + (−0.159 − 0.492i)14-s + (0.766 − 2.35i)15-s + (0.898 + 0.652i)16-s + (−0.506 − 0.367i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0531 + 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0531 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.106415 - 0.112234i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.106415 - 0.112234i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.16 - 6.65i)T \) |
| 11 | \( 1 + (-35.0 + 10.2i)T \) |
| good | 2 | \( 1 + (3.13 - 2.27i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (-2.57 - 7.91i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (14.0 + 10.1i)T + (38.6 + 118. i)T^{2} \) |
| 13 | \( 1 + (45.3 - 32.9i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (35.4 + 25.7i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (25.9 + 79.7i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 57.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-8.59 + 26.4i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (191. - 138. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (50.0 - 154. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (128. + 394. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 370.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-36.8 - 113. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (592. - 430. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-13.4 + 41.2i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-78.2 - 56.8i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-473. - 344. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (237. - 730. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-237. + 172. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-53.7 - 39.0i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-1.06e3 + 775. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−15.41411498090160853164055946856, −14.32253162139816590364577684561, −12.36771903194488913894866389341, −11.23564375457073909800471767071, −9.721096158383937075504641315518, −8.978934936224115995743216570284, −8.441692038979295027910239422097, −6.99233859912836971102761206261, −4.85609646084830604153063003916, −3.75771896104744378761659586281,
0.12881703621905561325705988766, 1.92554491348049557513604716698, 3.43336969383719791652177851037, 6.65100022926399824510063739865, 7.66244795450040737112179054517, 8.271579399798077821336516255912, 9.815523865397845702052453558143, 11.08810463210502592416277236338, 11.94916200347088088879157171016, 12.73554861693258412976948422146
