L(s) = 1 | + (−0.0611 − 0.228i)2-s + (0.965 + 0.258i)3-s + (1.68 − 0.972i)4-s + (1.04 − 1.97i)5-s − 0.236i·6-s + (−0.658 − 0.658i)8-s + (0.866 + 0.499i)9-s + (−0.515 − 0.117i)10-s + (−1.99 − 3.45i)11-s + (1.87 − 0.503i)12-s + (−0.500 + 0.500i)13-s + (1.52 − 1.63i)15-s + (1.83 − 3.17i)16-s + (0.614 − 2.29i)17-s + (0.0611 − 0.228i)18-s + (−3.60 + 6.25i)19-s + ⋯ |
L(s) = 1 | + (−0.0432 − 0.161i)2-s + (0.557 + 0.149i)3-s + (0.841 − 0.486i)4-s + (0.467 − 0.884i)5-s − 0.0964i·6-s + (−0.232 − 0.232i)8-s + (0.288 + 0.166i)9-s + (−0.162 − 0.0371i)10-s + (−0.600 − 1.04i)11-s + (0.542 − 0.145i)12-s + (−0.138 + 0.138i)13-s + (0.392 − 0.423i)15-s + (0.458 − 0.794i)16-s + (0.148 − 0.556i)17-s + (0.0144 − 0.0537i)18-s + (−0.828 + 1.43i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.82242 - 1.29857i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82242 - 1.29857i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.965 - 0.258i)T \) |
| 5 | \( 1 + (-1.04 + 1.97i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.0611 + 0.228i)T + (-1.73 + i)T^{2} \) |
| 11 | \( 1 + (1.99 + 3.45i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.500 - 0.500i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.614 + 2.29i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (3.60 - 6.25i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.04 + 1.88i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 3.65iT - 29T^{2} \) |
| 31 | \( 1 + (4.27 - 2.46i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.106 - 0.399i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 7.63iT - 41T^{2} \) |
| 43 | \( 1 + (-3.65 - 3.65i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.417 - 0.111i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.97 + 7.37i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.05 - 5.29i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.15 - 3.55i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.28 + 0.345i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 1.19T + 71T^{2} \) |
| 73 | \( 1 + (-1.88 - 0.506i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-7.48 - 4.32i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.9 + 11.9i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.91 + 6.77i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.43 + 7.43i)T + 97iT^{2} \) |
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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
−10.25896351686179184333657631658, −9.357992437234640225684677865934, −8.600281719964671384147819463498, −7.76382818890873047510902439903, −6.62759745063098631412269264528, −5.69165056350705128171257067209, −4.90767581925933277345207610642, −3.43015872580151572693701726448, −2.36759355192629916168651605412, −1.14791690118551016948437351597,
2.09990195003607746699638570706, 2.66633520587242778898720570119, 3.82574731302468326337874862082, 5.28016210804028446197931458846, 6.47931558962248240559015871127, 7.15161782317007432970677508548, 7.67430107228630524052951876501, 8.805968763168811241602026085063, 9.683146745001442630717460867800, 10.71527061739897197904060703637