L(s) = 1 | + (2.38 − 1.52i)2-s − 5.25·3-s + (3.36 − 7.25i)4-s − 19.3·5-s + (−12.5 + 7.98i)6-s + 22.0i·7-s + (−3.00 − 22.4i)8-s + 0.564·9-s + (−46.0 + 29.3i)10-s + 14.5i·11-s + (−17.6 + 38.0i)12-s − 75.1i·13-s + (33.6 + 52.6i)14-s + 101.·15-s + (−41.2 − 48.9i)16-s − 70.0·17-s + ⋯ |
L(s) = 1 | + (0.842 − 0.537i)2-s − 1.01·3-s + (0.421 − 0.906i)4-s − 1.72·5-s + (−0.851 + 0.543i)6-s + 1.19i·7-s + (−0.132 − 0.991i)8-s + 0.0209·9-s + (−1.45 + 0.929i)10-s + 0.397i·11-s + (−0.425 + 0.916i)12-s − 1.60i·13-s + (0.641 + 1.00i)14-s + 1.74·15-s + (−0.645 − 0.764i)16-s − 1.00·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0454503 + 0.254637i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0454503 + 0.254637i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.38 + 1.52i)T \) |
| 19 | \( 1 + (6.74 + 82.5i)T \) |
good | 3 | \( 1 + 5.25T + 27T^{2} \) |
| 5 | \( 1 + 19.3T + 125T^{2} \) |
| 7 | \( 1 - 22.0iT - 343T^{2} \) |
| 11 | \( 1 - 14.5iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 75.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 70.0T + 4.91e3T^{2} \) |
| 23 | \( 1 - 50.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 186. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 194.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 225. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 35.2iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 11.1iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 365. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 53.8iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 525.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 235.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 80.5T + 3.00e5T^{2} \) |
| 71 | \( 1 - 362.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 331.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 247. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.13e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 438. iT - 9.12e5T^{2} \) |
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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
−12.77640258419844344931622764439, −12.32409978217230660883727885047, −11.29731215626494002025681104514, −10.84366963667468333201077919428, −8.857248429978718621449123861270, −7.23434275848052691340058201768, −5.72082982733626960710503621337, −4.71097042579665613727793418411, −3.05642099588949188229357174667, −0.13661362123758977786946262197,
3.84512179654931133546833075406, 4.58095776110434744034976156462, 6.40236568238670193496509992278, 7.29332403723921815977858262270, 8.460033340347449379679654996462, 10.88442962532078444010916459470, 11.50994260776856687716107732462, 12.22072782712399125984527458322, 13.60590702769976027192132881695, 14.63113403854019742009197181201