L(s) = 1 | + (−2.57 − 2.57i)5-s + 3.74·7-s + (3.64 − 3.64i)11-s + (4.64 + 4.64i)13-s + 0.913i·17-s + (1.41 − 1.41i)19-s + 6i·23-s + 8.29i·25-s + (1.66 − 1.66i)29-s + 6.57i·31-s + (−9.64 − 9.64i)35-s + (1 − i)37-s − 0.913·41-s + (−3.74 − 3.74i)43-s + 6·47-s + ⋯ |
L(s) = 1 | + (−1.15 − 1.15i)5-s + 1.41·7-s + (1.09 − 1.09i)11-s + (1.28 + 1.28i)13-s + 0.221i·17-s + (0.324 − 0.324i)19-s + 1.25i·23-s + 1.65i·25-s + (0.309 − 0.309i)29-s + 1.18i·31-s + (−1.63 − 1.63i)35-s + (0.164 − 0.164i)37-s − 0.142·41-s + (−0.570 − 0.570i)43-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.009738977\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.009738977\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.57 + 2.57i)T + 5iT^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 + (-3.64 + 3.64i)T - 11iT^{2} \) |
| 13 | \( 1 + (-4.64 - 4.64i)T + 13iT^{2} \) |
| 17 | \( 1 - 0.913iT - 17T^{2} \) |
| 19 | \( 1 + (-1.41 + 1.41i)T - 19iT^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + (-1.66 + 1.66i)T - 29iT^{2} \) |
| 31 | \( 1 - 6.57iT - 31T^{2} \) |
| 37 | \( 1 + (-1 + i)T - 37iT^{2} \) |
| 41 | \( 1 + 0.913T + 41T^{2} \) |
| 43 | \( 1 + (3.74 + 3.74i)T + 43iT^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + (-3.49 - 3.49i)T + 53iT^{2} \) |
| 59 | \( 1 + (1.29 - 1.29i)T - 59iT^{2} \) |
| 61 | \( 1 + (0.291 + 0.291i)T + 61iT^{2} \) |
| 67 | \( 1 + (-3.32 + 3.32i)T - 67iT^{2} \) |
| 71 | \( 1 - 13.2iT - 71T^{2} \) |
| 73 | \( 1 + 8.58iT - 73T^{2} \) |
| 79 | \( 1 + 8.39iT - 79T^{2} \) |
| 83 | \( 1 + (-4.93 - 4.93i)T + 83iT^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.827814453190812192759774380237, −8.336216740601052721250887902958, −7.61742229078055543935843953842, −6.66448482537502666368836368777, −5.65377945847044564680691629737, −4.82151072011628469034698500956, −4.05492104602140819571845471563, −3.55490914519868632948305527112, −1.59160157075674811593525284436, −1.02563512935256381238991872892,
1.01871525002260575035601934293, 2.31439156532069832683450938952, 3.46280810175430962754071873495, 4.13041001687417385133310049751, 4.92706644529448064475458016131, 6.11477669780253194463639328701, 6.85288919413786625224943647374, 7.69360736261297845741883750165, 8.076150886547941785299837364798, 8.850481980914814692286425515982