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
−10.02847989252281458088335024695, −9.355126690405173109754772927817, −8.190936435389165630910651834622, −7.47969132659519591064697312697, −6.57147687967328299868833526205, −5.77847218492262364167986703892, −4.33959264596885262040194215604, −3.56747342623340460328252878158, −2.68922710630956101413725159302, −2.14010540079586024503031326743,
1.52378671386282863829721407126, 3.26773482731628491530108945789, 4.13861748031686010621137143926, 4.66809550123662252904540669483, 5.43155794105905768701660208822, 6.99068535293607896532002428088, 7.61840409151411350544663695916, 8.343313035810005419020289766339, 8.773381583329363817787498820014, 9.677308228539332767750085608456