L(s) = 1 | + (0.826 + 0.563i)3-s + (−0.988 + 0.149i)7-s + (0.365 + 0.930i)9-s + (1.03 + 1.29i)13-s + (0.988 − 1.71i)19-s + (−0.900 − 0.433i)21-s + (−0.988 − 0.149i)25-s + (−0.222 + 0.974i)27-s + (−0.955 − 1.65i)31-s + (−1.40 − 0.432i)37-s + (0.123 + 1.64i)39-s + (−0.658 − 0.317i)43-s + (0.955 − 0.294i)49-s + (1.78 − 0.858i)57-s + (−0.425 − 0.131i)61-s + ⋯ |
L(s) = 1 | + (0.826 + 0.563i)3-s + (−0.988 + 0.149i)7-s + (0.365 + 0.930i)9-s + (1.03 + 1.29i)13-s + (0.988 − 1.71i)19-s + (−0.900 − 0.433i)21-s + (−0.988 − 0.149i)25-s + (−0.222 + 0.974i)27-s + (−0.955 − 1.65i)31-s + (−1.40 − 0.432i)37-s + (0.123 + 1.64i)39-s + (−0.658 − 0.317i)43-s + (0.955 − 0.294i)49-s + (1.78 − 0.858i)57-s + (−0.425 − 0.131i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.103688507\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.103688507\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.826 - 0.563i)T \) |
| 7 | \( 1 + (0.988 - 0.149i)T \) |
good | 5 | \( 1 + (0.988 + 0.149i)T^{2} \) |
| 11 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 13 | \( 1 + (-1.03 - 1.29i)T + (-0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.0747 + 0.997i)T^{2} \) |
| 19 | \( 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.0747 - 0.997i)T^{2} \) |
| 29 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + (0.955 + 1.65i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (1.40 + 0.432i)T + (0.826 + 0.563i)T^{2} \) |
| 41 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 43 | \( 1 + (0.658 + 0.317i)T + (0.623 + 0.781i)T^{2} \) |
| 47 | \( 1 + (-0.955 + 0.294i)T^{2} \) |
| 53 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 59 | \( 1 + (0.988 - 0.149i)T^{2} \) |
| 61 | \( 1 + (0.425 + 0.131i)T + (0.826 + 0.563i)T^{2} \) |
| 67 | \( 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 73 | \( 1 + (-1.44 - 0.218i)T + (0.955 + 0.294i)T^{2} \) |
| 79 | \( 1 + (0.955 - 1.65i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 89 | \( 1 + (0.733 - 0.680i)T^{2} \) |
| 97 | \( 1 - 1.24T + T^{2} \) |
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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
−10.99246476794342574126043665573, −9.835386640828693563790628829492, −9.276748921734576988174056039974, −8.687492831505999117808719766486, −7.45379832682050987655875903231, −6.62387183834305951477772957372, −5.43589810531843183286277780432, −4.16217554851176583907011092286, −3.38449370565176405325727462572, −2.14655873873253757769721274675,
1.51278655266175496389788403266, 3.28239734147959649140593305471, 3.57514782796243959163573400822, 5.51539921416181138930919721621, 6.34772482385799728347617381128, 7.37643067881230184360989632866, 8.130284454149512787868117493493, 8.966024741901565362504048209400, 9.940223902234651432235611204604, 10.52690946378409646825227710245