L(s) = 1 | + (0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s − i·5-s + (0.866 + 0.499i)6-s + (−0.866 − 0.5i)7-s + i·8-s + (−0.5 − 0.866i)10-s + (0.866 − 0.5i)11-s + 13-s − 0.999·14-s + (0.866 − 0.5i)15-s + (0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s − 0.999i·21-s + (0.499 − 0.866i)22-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s − i·5-s + (0.866 + 0.499i)6-s + (−0.866 − 0.5i)7-s + i·8-s + (−0.5 − 0.866i)10-s + (0.866 − 0.5i)11-s + 13-s − 0.999·14-s + (0.866 − 0.5i)15-s + (0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s − 0.999i·21-s + (0.499 − 0.866i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.036534073\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.036534073\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + iT \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 + 2T + T^{2} \) |
| 89 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
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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
−9.222156917677372852900537656536, −8.793036275765040021888529734696, −8.058855341007834058645326891460, −6.75040124717050228992262232807, −5.90286168760770248318018196157, −4.94760572378684714890200114083, −4.05807006694459404990572403087, −3.75365517112442613391113443739, −2.98368547026787120932660667826, −1.35697326076553220875024877427,
1.50933507309928468738952462602, 2.78094736662485428386819332448, 3.49073516583723840525631366810, 4.46081993106333233479484009585, 5.64115311147651199572814929322, 6.35395400830920339656943634574, 6.93508351685398489942743926963, 7.30247933397961284366346906580, 8.565900037316766298991033388829, 9.251870867063070406454399412845