| L(s) = 1 | + (0.546 − 0.397i)2-s + (0.309 − 0.951i)3-s + (−0.476 + 1.46i)4-s + (−2.15 − 0.602i)5-s + (−0.208 − 0.642i)6-s − 7-s + (0.739 + 2.27i)8-s + (−0.809 − 0.587i)9-s + (−1.41 + 0.525i)10-s + (−2.81 + 2.04i)11-s + (1.24 + 0.907i)12-s + (−3.95 − 2.87i)13-s + (−0.546 + 0.397i)14-s + (−1.23 + 1.86i)15-s + (−1.18 − 0.863i)16-s + (−2.03 − 6.26i)17-s + ⋯ |
| L(s) = 1 | + (0.386 − 0.280i)2-s + (0.178 − 0.549i)3-s + (−0.238 + 0.733i)4-s + (−0.963 − 0.269i)5-s + (−0.0852 − 0.262i)6-s − 0.377·7-s + (0.261 + 0.805i)8-s + (−0.269 − 0.195i)9-s + (−0.447 + 0.166i)10-s + (−0.849 + 0.617i)11-s + (0.360 + 0.261i)12-s + (−1.09 − 0.797i)13-s + (−0.146 + 0.106i)14-s + (−0.319 + 0.480i)15-s + (−0.297 − 0.215i)16-s + (−0.493 − 1.51i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 - 0.407i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00552389 + 0.0259133i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00552389 + 0.0259133i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (2.15 + 0.602i)T \) |
| 7 | \( 1 + T \) |
| good | 2 | \( 1 + (-0.546 + 0.397i)T + (0.618 - 1.90i)T^{2} \) |
| 11 | \( 1 + (2.81 - 2.04i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (3.95 + 2.87i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (2.03 + 6.26i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.575 - 1.77i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (7.47 - 5.43i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.16 + 3.57i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.17 - 9.78i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.499 + 0.362i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.40 + 1.01i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 5.61T + 43T^{2} \) |
| 47 | \( 1 + (-2.50 + 7.71i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (0.826 - 2.54i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.32 - 0.961i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.19 - 1.59i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (2.63 + 8.11i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.65 + 8.15i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (5.16 - 3.74i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.09 - 6.44i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (2.16 + 6.65i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (2.88 - 2.09i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.96 + 12.1i)T + (-78.4 - 57.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.75422419456157155317483482332, −10.43407232946921346522504361142, −9.413682162135388838586408059175, −8.308277515274534118497521775039, −7.60768918298973765502628283228, −7.14254735841415849207022883677, −5.37072578497572034190590797517, −4.54306154222022528437147676510, −3.33235598281604009615530936430, −2.45247575662323658286778427144,
0.01253418592694562279925853762, 2.52441586142080998583120157173, 4.03085987369198800230826099152, 4.52498581677690945537713782087, 5.81263378774952560178097657865, 6.64271662334510013864365469855, 7.80307839866601074473136542048, 8.671948904100685447018071257649, 9.754355573022499454758726794022, 10.45275859484492145296155006690
