L(s) = 1 | + (−0.734 − 0.533i)2-s + (−0.0966 + 0.297i)3-s + (−0.0542 − 0.166i)4-s + (0.809 − 0.587i)5-s + (0.229 − 0.166i)6-s + (−0.329 + 1.01i)8-s + (0.729 + 0.530i)9-s − 0.907·10-s + (−0.587 + 0.809i)11-s + 0.0549·12-s + (1.44 + 1.04i)13-s + (0.0966 + 0.297i)15-s + (0.642 − 0.466i)16-s + (−0.253 − 0.779i)18-s + (−0.309 + 0.951i)19-s + (−0.142 − 0.103i)20-s + ⋯ |
L(s) = 1 | + (−0.734 − 0.533i)2-s + (−0.0966 + 0.297i)3-s + (−0.0542 − 0.166i)4-s + (0.809 − 0.587i)5-s + (0.229 − 0.166i)6-s + (−0.329 + 1.01i)8-s + (0.729 + 0.530i)9-s − 0.907·10-s + (−0.587 + 0.809i)11-s + 0.0549·12-s + (1.44 + 1.04i)13-s + (0.0966 + 0.297i)15-s + (0.642 − 0.466i)16-s + (−0.253 − 0.779i)18-s + (−0.309 + 0.951i)19-s + (−0.142 − 0.103i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7903186761\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7903186761\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (0.587 - 0.809i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
good | 2 | \( 1 + (0.734 + 0.533i)T + (0.309 + 0.951i)T^{2} \) |
| 3 | \( 1 + (0.0966 - 0.297i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-1.44 - 1.04i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.610 + 1.87i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (1.14 + 0.831i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 67 | \( 1 - 1.97T + T^{2} \) |
| 71 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.734 - 0.533i)T + (0.309 + 0.951i)T^{2} \) |
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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
−9.939993420244897079282035905580, −9.507480821109241928820282336887, −8.680637887175307277620337005796, −7.935881902001804788983198218211, −6.65750274004771855022760676568, −5.69144289670829684480951867706, −4.92039563256432270801879614304, −3.95013229823751754035397440149, −2.10352342462384947649298661910, −1.54725942861905567362802865192,
1.12502795633067642460362405588, 2.90558334796550583097059412773, 3.71401083240075160954648650323, 5.28595692428633137329763899387, 6.39070970102582172174071527902, 6.63896771509090242549859827856, 7.77601105681662891255822514548, 8.429361754337059697065156766435, 9.219731382167775391335486715281, 10.08024534562062616925400232620