L(s) = 1 | + (−1.53 + 1.11i)2-s + (−0.363 − 1.11i)3-s + (0.809 − 2.48i)4-s + (−0.809 − 0.587i)5-s + (1.80 + 1.31i)6-s + (0.951 + 2.92i)8-s + (−0.309 + 0.224i)9-s + 1.90·10-s + (0.809 − 0.587i)11-s − 3.07·12-s + (1.53 − 1.11i)13-s + (−0.363 + 1.11i)15-s + (−2.61 − 1.90i)16-s + (0.224 − 0.690i)18-s + (0.309 + 0.951i)19-s + (−2.11 + 1.53i)20-s + ⋯ |
L(s) = 1 | + (−1.53 + 1.11i)2-s + (−0.363 − 1.11i)3-s + (0.809 − 2.48i)4-s + (−0.809 − 0.587i)5-s + (1.80 + 1.31i)6-s + (0.951 + 2.92i)8-s + (−0.309 + 0.224i)9-s + 1.90·10-s + (0.809 − 0.587i)11-s − 3.07·12-s + (1.53 − 1.11i)13-s + (−0.363 + 1.11i)15-s + (−2.61 − 1.90i)16-s + (0.224 − 0.690i)18-s + (0.309 + 0.951i)19-s + (−2.11 + 1.53i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4024669212\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4024669212\) |
\(L(1)\) |
|
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 | \( 1 + (-0.809 + 0.587i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 3 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 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 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - 1.17T + T^{2} \) |
| 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 | \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \) |
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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
−9.600982630043050337213174288406, −8.783725013556343934957782211171, −8.072675605173069013063847476450, −7.74542040187738281264754439318, −6.73220540186359521013115819523, −6.08891112844761003354731306522, −5.42428041034082298199854303362, −3.71153123224264895993674475358, −1.51129745340735722523562639872, −0.77320405199323260281190916169,
1.49648414052574090114986320280, 3.00527518065222213294863655595, 3.90959356959646672118857638629, 4.46084979734364782089507129783, 6.46454936074099107833956501188, 7.16753131619397321580552380384, 8.183781501472548031765011332587, 9.011763855758932671481951408294, 9.519724862622821140198105343276, 10.32462275630242761523685896718