| L(s) = 1 | + (4.45 + 2.66i)3-s − 8.13i·5-s − 7i·7-s + (12.7 + 23.7i)9-s − 27.3·11-s − 45.1·13-s + (21.7 − 36.3i)15-s + 20.4i·17-s + 147. i·19-s + (18.6 − 31.2i)21-s − 81.6·23-s + 58.7·25-s + (−6.39 + 140. i)27-s + 76.2i·29-s + 214. i·31-s + ⋯ |
| L(s) = 1 | + (0.858 + 0.513i)3-s − 0.728i·5-s − 0.377i·7-s + (0.473 + 0.880i)9-s − 0.750·11-s − 0.963·13-s + (0.373 − 0.624i)15-s + 0.291i·17-s + 1.78i·19-s + (0.193 − 0.324i)21-s − 0.740·23-s + 0.469·25-s + (−0.0455 + 0.998i)27-s + 0.488i·29-s + 1.24i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.783614990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.783614990\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.45 - 2.66i)T \) |
| 7 | \( 1 + 7iT \) |
| good | 5 | \( 1 + 8.13iT - 125T^{2} \) |
| 11 | \( 1 + 27.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 147. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 81.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 76.2iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 214. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 175.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 298. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 110. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 306.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 156. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 717.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 296.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 289. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 880.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 175.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 454. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 806.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 559. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−10.09821032356006700894120596214, −9.652311092454794775048052869385, −8.453885248042513334947470655946, −8.042860108403811033338547681102, −7.07940962793040523959516769400, −5.60159205997425101373125640656, −4.74405225862333055022824177857, −3.85862496777931802330432204076, −2.69732155437738817264960088668, −1.43714343353940186326857857677,
0.43073538245193634882130077829, 2.42656174351393537863732305075, 2.64593143536164693298688401624, 4.10170793011399803986731903328, 5.33505410695270175429000716037, 6.56289725766606512646229673977, 7.29581701262010176906582276363, 7.973298801956500224374463734594, 9.036149295252026522261787309866, 9.698068251665556670164083835935
