L(s) = 1 | + (−0.809 − 0.587i)4-s + (−0.831 + 1.14i)7-s + (1.34 + 0.437i)13-s + (0.309 + 0.951i)16-s + (0.831 + 1.14i)19-s + (0.809 − 0.587i)25-s + (1.34 − 0.437i)28-s + 1.41i·43-s + (−0.309 − 0.951i)49-s + (−0.831 − 1.14i)52-s + (−1.34 + 0.437i)61-s + (0.309 − 0.951i)64-s + (0.831 − 1.14i)73-s − 1.41i·76-s + (−1.34 − 0.437i)79-s + ⋯ |
L(s) = 1 | + (−0.809 − 0.587i)4-s + (−0.831 + 1.14i)7-s + (1.34 + 0.437i)13-s + (0.309 + 0.951i)16-s + (0.831 + 1.14i)19-s + (0.809 − 0.587i)25-s + (1.34 − 0.437i)28-s + 1.41i·43-s + (−0.309 − 0.951i)49-s + (−0.831 − 1.14i)52-s + (−1.34 + 0.437i)61-s + (0.309 − 0.951i)64-s + (0.831 − 1.14i)73-s − 1.41i·76-s + (−1.34 − 0.437i)79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8069632794\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8069632794\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - 1.41iT - T^{2} \) |
| 47 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (1.34 + 0.437i)T + (0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
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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
−10.00296344056715117179984648397, −9.245410120655651657865548641000, −8.765036093098545803853242524841, −7.899477059972208865377789057462, −6.38835720984360471504549436386, −6.01776256090067706358635154855, −5.10852393856463258337694558068, −3.98516837862355863464745061497, −3.00389255426501251501802060626, −1.44427171151413130515009200183,
0.876610300552967539927896933030, 3.09317449994281244609463585816, 3.66785063532021729064990312177, 4.62295279559899667408415720097, 5.66884884602400640713793558031, 6.84927620247096804116122476174, 7.42066626237970297436982704260, 8.454730587934013441016343683432, 9.113872714005021449851708913365, 9.911954113452877835483747869360