L(s) = 1 | + (0.733 + 0.680i)2-s + (0.988 + 0.149i)3-s + (0.0747 + 0.997i)4-s + (−0.698 − 1.77i)5-s + (0.623 + 0.781i)6-s + (0.365 − 0.930i)7-s + (−0.623 + 0.781i)8-s + (0.955 + 0.294i)9-s + (0.698 − 1.77i)10-s + (−0.142 + 0.0440i)11-s + (−0.0747 + 0.997i)12-s + (0.900 − 0.433i)14-s + (−0.425 − 1.86i)15-s + (−0.988 + 0.149i)16-s + (0.5 + 0.866i)18-s + ⋯ |
L(s) = 1 | + (0.733 + 0.680i)2-s + (0.988 + 0.149i)3-s + (0.0747 + 0.997i)4-s + (−0.698 − 1.77i)5-s + (0.623 + 0.781i)6-s + (0.365 − 0.930i)7-s + (−0.623 + 0.781i)8-s + (0.955 + 0.294i)9-s + (0.698 − 1.77i)10-s + (−0.142 + 0.0440i)11-s + (−0.0747 + 0.997i)12-s + (0.900 − 0.433i)14-s + (−0.425 − 1.86i)15-s + (−0.988 + 0.149i)16-s + (0.5 + 0.866i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.904902808\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.904902808\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.733 - 0.680i)T \) |
| 3 | \( 1 + (-0.988 - 0.149i)T \) |
| 7 | \( 1 + (-0.365 + 0.930i)T \) |
good | 5 | \( 1 + (0.698 + 1.77i)T + (-0.733 + 0.680i)T^{2} \) |
| 11 | \( 1 + (0.142 - 0.0440i)T + (0.826 - 0.563i)T^{2} \) |
| 13 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 17 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 29 | \( 1 + (1.32 - 0.636i)T + (0.623 - 0.781i)T^{2} \) |
| 31 | \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 41 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 43 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 53 | \( 1 + (-0.0747 - 0.997i)T + (-0.988 + 0.149i)T^{2} \) |
| 59 | \( 1 + (-0.365 + 0.930i)T + (-0.733 - 0.680i)T^{2} \) |
| 61 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (0.914 - 0.848i)T + (0.0747 - 0.997i)T^{2} \) |
| 79 | \( 1 + (0.365 - 0.632i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.326 + 1.42i)T + (-0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 97 | \( 1 - 1.65T + T^{2} \) |
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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
−9.677308228539332767750085608456, −8.773381583329363817787498820014, −8.343313035810005419020289766339, −7.61840409151411350544663695916, −6.99068535293607896532002428088, −5.43155794105905768701660208822, −4.66809550123662252904540669483, −4.13861748031686010621137143926, −3.26773482731628491530108945789, −1.52378671386282863829721407126,
2.14010540079586024503031326743, 2.68922710630956101413725159302, 3.56747342623340460328252878158, 4.33959264596885262040194215604, 5.77847218492262364167986703892, 6.57147687967328299868833526205, 7.47969132659519591064697312697, 8.190936435389165630910651834622, 9.355126690405173109754772927817, 10.02847989252281458088335024695