L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + (0.499 − 0.866i)12-s + 0.999·15-s + (−0.499 + 0.866i)16-s + (−1 − 1.73i)17-s + (0.5 − 0.866i)19-s − 0.999·20-s + 27-s − 29-s − 41-s − 0.999·48-s + (0.999 − 1.73i)51-s + (0.5 + 0.866i)53-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + (0.499 − 0.866i)12-s + 0.999·15-s + (−0.499 + 0.866i)16-s + (−1 − 1.73i)17-s + (0.5 − 0.866i)19-s − 0.999·20-s + 27-s − 29-s − 41-s − 0.999·48-s + (0.999 − 1.73i)51-s + (0.5 + 0.866i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.320521024\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.320521024\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 2T + T^{2} \) |
| 73 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.151552720672709293823699975324, −8.502879611332539410857679956041, −7.25128782032203409565381611584, −6.46750471128710390688533075515, −5.35434777230991222905767567092, −4.93845402367619572127930918241, −4.33709686595721967412685829130, −3.25189859017203059997583315054, −2.09395922604873230202677430274, −0.799467565213600756285561548217,
1.72631009811630911762167908708, 2.44134576684263061045762235445, 3.42596919566228130836220305286, 4.12937392474560267474951653692, 5.30286605319036745147130926708, 6.35369637138782202490704415144, 6.88335806846954413555967232607, 7.68672783402161022958912402917, 8.239776327952519283819585513749, 8.823573607109316093543880801994