L(s) = 1 | + (−1.36 − 0.366i)2-s + (0.684 + 2.55i)3-s + (1.73 + i)4-s + (2.23 − 0.133i)5-s − 3.74i·6-s + (−2.55 + 0.684i)7-s + (−1.99 − 2i)8-s + (−3.46 + 2i)9-s + (−3.09 − 0.633i)10-s + (3.24 + 1.87i)11-s + (−1.36 + 5.11i)12-s + (−2 − 2i)13-s + 3.74·14-s + (1.87 + 5.61i)15-s + (1.99 + 3.46i)16-s + (0.732 + 2.73i)17-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.395 + 1.47i)3-s + (0.866 + 0.5i)4-s + (0.998 − 0.0599i)5-s − 1.52i·6-s + (−0.965 + 0.258i)7-s + (−0.707 − 0.707i)8-s + (−1.15 + 0.666i)9-s + (−0.979 − 0.200i)10-s + (0.977 + 0.564i)11-s + (−0.395 + 1.47i)12-s + (−0.554 − 0.554i)13-s + 0.999·14-s + (0.483 + 1.44i)15-s + (0.499 + 0.866i)16-s + (0.177 + 0.662i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.709030 + 0.525482i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.709030 + 0.525482i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.36 + 0.366i)T \) |
| 5 | \( 1 + (-2.23 + 0.133i)T \) |
| 7 | \( 1 + (2.55 - 0.684i)T \) |
good | 3 | \( 1 + (-0.684 - 2.55i)T + (-2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (-3.24 - 1.87i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2 + 2i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.732 - 2.73i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.87 + 3.24i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.55 - 0.684i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 3iT - 29T^{2} \) |
| 31 | \( 1 + (6.48 + 3.74i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + (-5.61 + 5.61i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.73 + 10.2i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-6.83 + 1.83i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (1.87 - 3.24i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 - 2.59i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (12.7 - 3.42i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 3.74iT - 71T^{2} \) |
| 73 | \( 1 + (2.73 - 0.732i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.87 - 3.24i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.87 - 1.87i)T - 83iT^{2} \) |
| 89 | \( 1 + (-2.59 + 1.5i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9 - 9i)T - 97iT^{2} \) |
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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
−13.25070966040520212892619000289, −12.21372132215837403933814314200, −10.75134636571187914034512693168, −10.04678113800125763124540710689, −9.339932143018132992056068766123, −8.838100821170792775595441961361, −7.00433647450526368200013376986, −5.69355420092330337635434911839, −3.90244369055853956568921339215, −2.52599602846283328643248464874,
1.36060225857218438043513669614, 2.79243026387659513564798109453, 5.94137313022502004013208214074, 6.69376241350682972458296218848, 7.43140810436718105673882173618, 8.894743760306749312137481492597, 9.468328193613773758388099228737, 10.75750357625878724601687080143, 12.10671777155955384707011941601, 12.91207071533479436412073410489