L(s) = 1 | − 2·3-s − 3.46i·5-s + (−2 − 1.73i)7-s + 9-s + 3.46i·11-s + 3.46i·13-s + 6.92i·15-s − 2·19-s + (4 + 3.46i)21-s + 3.46i·23-s − 6.99·25-s + 4·27-s − 6·29-s − 8·31-s − 6.92i·33-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1.54i·5-s + (−0.755 − 0.654i)7-s + 0.333·9-s + 1.04i·11-s + 0.960i·13-s + 1.78i·15-s − 0.458·19-s + (0.872 + 0.755i)21-s + 0.722i·23-s − 1.39·25-s + 0.769·27-s − 1.11·29-s − 1.43·31-s − 1.20i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (2 + 1.73i)T \) |
good | 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 3.46iT - 67T^{2} \) |
| 71 | \( 1 + 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−10.62971877371711793557146197994, −9.543758927900699733677403390336, −9.042423861524244005037461886839, −7.63741190824055896911444704874, −6.69761540473636546177247356568, −5.66564929462574484923420618607, −4.79405130769669519368537773279, −3.95877480960822565011520621456, −1.60255508818230503477201738909, 0,
2.66407455232218199344391484644, 3.55831391959659470619351593443, 5.40773577478471368199510590756, 6.08787052147051313098771959178, 6.66180970489436224321061865150, 7.82500027110748167675903211301, 9.075988472235495468184945208390, 10.21870597245114522757813265230, 10.94336339895565410128113947485