| L(s) = 1 | + (−5.19 + 0.133i)3-s + (7.81 + 13.5i)5-s + (6.71 + 17.2i)7-s + (26.9 − 1.39i)9-s + (0.495 − 0.858i)11-s + (16.4 − 28.4i)13-s + (−42.4 − 69.2i)15-s + (−37.8 − 65.5i)17-s + (−62.5 + 108. i)19-s + (−37.1 − 88.7i)21-s + (104. + 181. i)23-s + (−59.6 + 103. i)25-s + (−139. + 10.8i)27-s + (−2.40 − 4.16i)29-s + 4.45·31-s + ⋯ |
| L(s) = 1 | + (−0.999 + 0.0257i)3-s + (0.698 + 1.21i)5-s + (0.362 + 0.931i)7-s + (0.998 − 0.0515i)9-s + (0.0135 − 0.0235i)11-s + (0.350 − 0.607i)13-s + (−0.729 − 1.19i)15-s + (−0.540 − 0.935i)17-s + (−0.754 + 1.30i)19-s + (−0.386 − 0.922i)21-s + (0.950 + 1.64i)23-s + (−0.476 + 0.825i)25-s + (−0.997 + 0.0772i)27-s + (−0.0153 − 0.0266i)29-s + 0.0257·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 - 0.789i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.205979042\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.205979042\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (5.19 - 0.133i)T \) |
| 7 | \( 1 + (-6.71 - 17.2i)T \) |
| good | 5 | \( 1 + (-7.81 - 13.5i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-0.495 + 0.858i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-16.4 + 28.4i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (37.8 + 65.5i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (62.5 - 108. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-104. - 181. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (2.40 + 4.16i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 4.45T + 2.97e4T^{2} \) |
| 37 | \( 1 + (104. - 181. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-10.3 + 17.9i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-103. - 179. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + 387.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (251. + 435. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + 489.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 80.7T + 2.26e5T^{2} \) |
| 67 | \( 1 + 32.0T + 3.00e5T^{2} \) |
| 71 | \( 1 + 704.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (5.26 + 9.12i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 - 1.31e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (77.8 + 134. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-98.1 + 170. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-859. - 1.48e3i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.67653638791914223715917358742, −11.06575586975283526755667744633, −10.20423164067949085964564788771, −9.320763772936340648704382793801, −7.84143528194990984501680630797, −6.65920511453922672280838765909, −5.92280068940122367447239297708, −5.01787537292938275948758045105, −3.21168649128881127217369161841, −1.74794747003270040886218161370,
0.55319640521141205332758866850, 1.74103644965988845083991561046, 4.34186123088914300462749098843, 4.84720836469360624898591066087, 6.17345062102767283896817137843, 7.00324136908719554274411342404, 8.496749217541212069203455962418, 9.315301366504043684723662789747, 10.61336084927607492062914349012, 11.02806487283189881032149626916
