L(s) = 1 | + (3.79 + 1.25i)2-s + (3.67 − 3.67i)3-s + (12.8 + 9.53i)4-s + (27.4 − 27.4i)5-s + (18.5 − 9.34i)6-s − 74.9·7-s + (36.8 + 52.3i)8-s − 27i·9-s + (138. − 69.8i)10-s + (107. + 107. i)11-s + (82.2 − 12.1i)12-s + (36.5 + 36.5i)13-s + (−284. − 94.0i)14-s − 201. i·15-s + (74.2 + 244. i)16-s − 343.·17-s + ⋯ |
L(s) = 1 | + (0.949 + 0.313i)2-s + (0.408 − 0.408i)3-s + (0.803 + 0.595i)4-s + (1.09 − 1.09i)5-s + (0.515 − 0.259i)6-s − 1.53·7-s + (0.575 + 0.817i)8-s − 0.333i·9-s + (1.38 − 0.698i)10-s + (0.886 + 0.886i)11-s + (0.571 − 0.0846i)12-s + (0.216 + 0.216i)13-s + (−1.45 − 0.480i)14-s − 0.897i·15-s + (0.290 + 0.957i)16-s − 1.18·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0982i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.995 + 0.0982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.98671 - 0.147092i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.98671 - 0.147092i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.79 - 1.25i)T \) |
| 3 | \( 1 + (-3.67 + 3.67i)T \) |
good | 5 | \( 1 + (-27.4 + 27.4i)T - 625iT^{2} \) |
| 7 | \( 1 + 74.9T + 2.40e3T^{2} \) |
| 11 | \( 1 + (-107. - 107. i)T + 1.46e4iT^{2} \) |
| 13 | \( 1 + (-36.5 - 36.5i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + 343.T + 8.35e4T^{2} \) |
| 19 | \( 1 + (102. - 102. i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 + 436.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (535. + 535. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 - 769. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (-1.69e3 + 1.69e3i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 1.69e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (1.13e3 + 1.13e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 979. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (-2.63e3 + 2.63e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + (-4.78e3 - 4.78e3i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (415. + 415. i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 + (-62.1 + 62.1i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 + 1.21e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 3.71e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 9.37e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (2.90e3 - 2.90e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 4.49e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.23e4T + 8.85e7T^{2} \) |
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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
−14.67401557104043529047354487647, −13.37186538605896019049597331440, −13.03148615642957591498838901713, −12.01811539181386342225209020544, −9.841510071973127809546200206003, −8.815137042653121203612443536820, −6.85877826428708519609554493180, −5.93782613921887748301286530027, −4.11994543221557853437233458105, −2.07672162635362361345663986941,
2.55006108098987947864789408236, 3.70216510137155053153384537254, 6.02660680983604352284622915701, 6.67014045980801794072029219919, 9.324173440710981847723089790404, 10.24135982706171485701109647954, 11.29697228292041621528707292678, 13.09119101320204043608762127171, 13.69302233603059240591008014753, 14.68672724156946734421956110076