L(s) = 1 | + (1.50 + 1.26i)3-s + (0.500 + 2.83i)9-s + (−1.32 − 0.766i)11-s + (−0.173 + 0.984i)17-s + (0.342 − 0.939i)19-s + (0.766 − 0.642i)25-s + (−1.85 + 3.20i)27-s + (−1.03 − 2.83i)33-s + (0.439 − 0.524i)41-s + (−0.342 − 0.939i)43-s + (0.5 − 0.866i)49-s + (−1.50 + 1.26i)51-s + (1.70 − 0.984i)57-s + (0.223 − 1.26i)59-s + (0.118 + 0.673i)67-s + ⋯ |
L(s) = 1 | + (1.50 + 1.26i)3-s + (0.500 + 2.83i)9-s + (−1.32 − 0.766i)11-s + (−0.173 + 0.984i)17-s + (0.342 − 0.939i)19-s + (0.766 − 0.642i)25-s + (−1.85 + 3.20i)27-s + (−1.03 − 2.83i)33-s + (0.439 − 0.524i)41-s + (−0.342 − 0.939i)43-s + (0.5 − 0.866i)49-s + (−1.50 + 1.26i)51-s + (1.70 − 0.984i)57-s + (0.223 − 1.26i)59-s + (0.118 + 0.673i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 - 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.614957541\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.614957541\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-0.342 + 0.939i)T \) |
good | 3 | \( 1 + (-1.50 - 1.26i)T + (0.173 + 0.984i)T^{2} \) |
| 5 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (1.32 + 0.766i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 23 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 29 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.439 + 0.524i)T + (-0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (0.342 + 0.939i)T + (-0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 53 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (-0.223 + 1.26i)T + (-0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.118 - 0.673i)T + (-0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 73 | \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (1.62 - 0.939i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-1.11 - 1.32i)T + (-0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (0.673 + 0.118i)T + (0.939 + 0.342i)T^{2} \) |
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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
−10.12650103303459200173593646919, −9.170161431816445025365527892474, −8.505790989733155251017273726893, −8.064691981213460502998996268523, −7.07248088703180243796092937104, −5.56481680083589258065144726303, −4.82324116115159638912987864850, −3.87506060054537706113741383488, −3.00960700019456537680335124638, −2.25824319032103595698018379715,
1.40818187545204399092170875035, 2.55060685991914191454508305256, 3.12674812637773752249467569328, 4.43574087110463525070322506297, 5.70749032343215114624482718433, 6.84002411968269996806298607909, 7.49733981940065841984371437629, 7.934046547069480907024325174019, 8.838494421297016767689388903521, 9.541387056712598894500457505417