L(s) = 1 | + (2.96 + 2.96i)5-s − 7-s + (0.569 − 0.569i)11-s + (2.53 + 2.53i)13-s − 1.03i·17-s + (5.23 − 5.23i)19-s + 8.71i·23-s + 12.6i·25-s + (−6.05 + 6.05i)29-s − 3.00i·31-s + (−2.96 − 2.96i)35-s + (−0.149 + 0.149i)37-s + 8.63·41-s + (−1.73 − 1.73i)43-s − 4.10·47-s + ⋯ |
L(s) = 1 | + (1.32 + 1.32i)5-s − 0.377·7-s + (0.171 − 0.171i)11-s + (0.703 + 0.703i)13-s − 0.251i·17-s + (1.20 − 1.20i)19-s + 1.81i·23-s + 2.52i·25-s + (−1.12 + 1.12i)29-s − 0.539i·31-s + (−0.501 − 0.501i)35-s + (−0.0246 + 0.0246i)37-s + 1.34·41-s + (−0.264 − 0.264i)43-s − 0.599·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0595 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0595 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.479665472\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.479665472\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (-2.96 - 2.96i)T + 5iT^{2} \) |
| 11 | \( 1 + (-0.569 + 0.569i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2.53 - 2.53i)T + 13iT^{2} \) |
| 17 | \( 1 + 1.03iT - 17T^{2} \) |
| 19 | \( 1 + (-5.23 + 5.23i)T - 19iT^{2} \) |
| 23 | \( 1 - 8.71iT - 23T^{2} \) |
| 29 | \( 1 + (6.05 - 6.05i)T - 29iT^{2} \) |
| 31 | \( 1 + 3.00iT - 31T^{2} \) |
| 37 | \( 1 + (0.149 - 0.149i)T - 37iT^{2} \) |
| 41 | \( 1 - 8.63T + 41T^{2} \) |
| 43 | \( 1 + (1.73 + 1.73i)T + 43iT^{2} \) |
| 47 | \( 1 + 4.10T + 47T^{2} \) |
| 53 | \( 1 + (6.04 + 6.04i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.05 + 6.05i)T - 59iT^{2} \) |
| 61 | \( 1 + (-5.81 - 5.81i)T + 61iT^{2} \) |
| 67 | \( 1 + (0.0256 - 0.0256i)T - 67iT^{2} \) |
| 71 | \( 1 - 14.5iT - 71T^{2} \) |
| 73 | \( 1 + 5.38iT - 73T^{2} \) |
| 79 | \( 1 + 3.89iT - 79T^{2} \) |
| 83 | \( 1 + (-1.40 - 1.40i)T + 83iT^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 - 3.75T + 97T^{2} \) |
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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
−8.929206963656740255158682950200, −7.59039233971022537582880070970, −7.03609558021018807891280046458, −6.49174479524176819595453077091, −5.69131659202179536902388068681, −5.19580265228937361202741782912, −3.71117844406687281639294852727, −3.17548094817171199018396452611, −2.25873286799453459345344994196, −1.34025784625278318802322132160,
0.73813084592862037776925907194, 1.59835066705095581004627105450, 2.56390135653085401397135724491, 3.71427883838953405441804139556, 4.59408295816790704443371184897, 5.42832706234068867695772058175, 5.96149745876881219218224803211, 6.49238958761718726547533713679, 7.78283595770235367676333053420, 8.306059196704966634690050651805