L(s) = 1 | + 1.58·2-s + (1.43 − 2.48i)3-s + 0.496·4-s + (1.71 + 2.97i)5-s + (2.26 − 3.92i)6-s + (0.908 − 1.57i)7-s − 2.37·8-s + (−2.62 − 4.54i)9-s + (2.71 + 4.70i)10-s + (0.701 + 1.21i)11-s + (0.712 − 1.23i)12-s + (0.5 + 0.866i)13-s + (1.43 − 2.48i)14-s + 9.86·15-s − 4.74·16-s + (0.332 − 0.576i)17-s + ⋯ |
L(s) = 1 | + 1.11·2-s + (0.828 − 1.43i)3-s + 0.248·4-s + (0.768 + 1.33i)5-s + (0.926 − 1.60i)6-s + (0.343 − 0.594i)7-s − 0.839·8-s + (−0.873 − 1.51i)9-s + (0.858 + 1.48i)10-s + (0.211 + 0.366i)11-s + (0.205 − 0.356i)12-s + (0.138 + 0.240i)13-s + (0.383 − 0.664i)14-s + 2.54·15-s − 1.18·16-s + (0.0807 − 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.79614 - 1.15937i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.79614 - 1.15937i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.769 - 5.51i)T \) |
good | 2 | \( 1 - 1.58T + 2T^{2} \) |
| 3 | \( 1 + (-1.43 + 2.48i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.71 - 2.97i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.908 + 1.57i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.701 - 1.21i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.332 + 0.576i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.934 + 1.61i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 6.53T + 23T^{2} \) |
| 29 | \( 1 + 5.25T + 29T^{2} \) |
| 37 | \( 1 + (4.05 - 7.01i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.91 + 3.30i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.255 + 0.441i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 9.00T + 47T^{2} \) |
| 53 | \( 1 + (3.50 + 6.07i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.41 + 9.37i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 7.62T + 61T^{2} \) |
| 67 | \( 1 + (-7.31 - 12.6i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.40 + 2.42i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.03 + 12.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.28 - 10.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.888 + 1.53i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 9.70T + 89T^{2} \) |
| 97 | \( 1 + 9.81T + 97T^{2} \) |
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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
−11.52647185565348374219991723873, −10.32582591547807077095633539092, −9.266781903479538966004934928590, −8.114845417109852325344814481095, −6.98817741348219252696316047154, −6.64188582409603158112437217969, −5.52586700635781570805729262584, −3.89356154615563363273307335717, −2.86770515618758385053149817576, −1.87474805995938378242087766842,
2.28796063927902972165345506798, 3.72331627183805244740231421045, 4.38726912621781812412006983237, 5.46817637648304453793201864816, 5.76235629302828386857113966561, 8.124290194355263139683189249040, 8.905945326285863623970112327970, 9.365294023070134790206207113520, 10.25150434249080392543796439567, 11.57698686717235087705444410930