L(s) = 1 | − 3-s + 3.38·5-s + (1.69 + 2.93i)7-s + 9-s + (1 + 1.73i)11-s + (1.53 − 2.65i)13-s − 3.38·15-s + (−1.03 + 1.78i)17-s + (−3.72 + 6.45i)19-s + (−1.69 − 2.93i)21-s + (1.11 − 1.92i)23-s + 6.44·25-s − 27-s + (−2.80 − 4.85i)29-s + (−2.99 − 5.18i)31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.51·5-s + (0.639 + 1.10i)7-s + 0.333·9-s + (0.301 + 0.522i)11-s + (0.425 − 0.736i)13-s − 0.873·15-s + (−0.250 + 0.433i)17-s + (−0.854 + 1.47i)19-s + (−0.369 − 0.639i)21-s + (0.231 − 0.401i)23-s + 1.28·25-s − 0.192·27-s + (−0.520 − 0.901i)29-s + (−0.537 − 0.931i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.67553 + 0.636258i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.67553 + 0.636258i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 67 | \( 1 + (7.80 - 2.46i)T \) |
good | 5 | \( 1 - 3.38T + 5T^{2} \) |
| 7 | \( 1 + (-1.69 - 2.93i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.53 + 2.65i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.03 - 1.78i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.72 - 6.45i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.11 + 1.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.80 + 4.85i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.99 + 5.18i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.14 - 5.44i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.92 - 3.32i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 5.92T + 43T^{2} \) |
| 47 | \( 1 + (-1.92 - 3.32i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 1.93T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + (-4.98 + 8.63i)T + (-30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (-6.65 - 11.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.80 + 4.85i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.29 + 10.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.03 - 3.52i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 0.837T + 89T^{2} \) |
| 97 | \( 1 + (-3.08 + 5.33i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.27984006414596316050924036735, −9.655083046063271155137123561018, −8.739138674499467502975916768821, −7.925343345848199609060117330142, −6.50472597271883575971069106001, −5.84707238469448003540157817085, −5.38318519546251203009101496077, −4.14012478527300088394886741299, −2.38898790673550169901712074357, −1.61724318612866070566807204850,
1.07543074553081861579739367971, 2.20601558774055147003288484817, 3.88209008676129061588437495665, 4.92870265848054464743164173312, 5.70628928602235235954802316651, 6.75812415467327417917281660210, 7.19231452048336106549494591254, 8.815634569396388427500147974383, 9.223471375414101183192512448697, 10.40875435245283119179744374133