L(s) = 1 | + (−1.16 − 1.16i)5-s + 3.74·7-s + (−1.64 + 1.64i)11-s + (0.645 + 0.645i)13-s − 6.57i·17-s + (1.41 − 1.41i)19-s − 6i·23-s − 2.29i·25-s + (−5.40 + 5.40i)29-s + 0.913i·31-s + (−4.35 − 4.35i)35-s + (−1 + i)37-s + 6.57·41-s + (3.74 + 3.74i)43-s − 6·47-s + ⋯ |
L(s) = 1 | + (−0.520 − 0.520i)5-s + 1.41·7-s + (−0.496 + 0.496i)11-s + (0.179 + 0.179i)13-s − 1.59i·17-s + (0.324 − 0.324i)19-s − 1.25i·23-s − 0.458i·25-s + (−1.00 + 1.00i)29-s + 0.164i·31-s + (−0.736 − 0.736i)35-s + (−0.164 + 0.164i)37-s + 1.02·41-s + (0.570 + 0.570i)43-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.648259146\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.648259146\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.16 + 1.16i)T + 5iT^{2} \) |
| 7 | \( 1 - 3.74T + 7T^{2} \) |
| 11 | \( 1 + (1.64 - 1.64i)T - 11iT^{2} \) |
| 13 | \( 1 + (-0.645 - 0.645i)T + 13iT^{2} \) |
| 17 | \( 1 + 6.57iT - 17T^{2} \) |
| 19 | \( 1 + (-1.41 + 1.41i)T - 19iT^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + (5.40 - 5.40i)T - 29iT^{2} \) |
| 31 | \( 1 - 0.913iT - 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 - 6.57T + 41T^{2} \) |
| 43 | \( 1 + (-3.74 - 3.74i)T + 43iT^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + (-7.73 - 7.73i)T + 53iT^{2} \) |
| 59 | \( 1 + (-9.29 + 9.29i)T - 59iT^{2} \) |
| 61 | \( 1 + (10.2 + 10.2i)T + 61iT^{2} \) |
| 67 | \( 1 + (-10.8 + 10.8i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.70iT - 71T^{2} \) |
| 73 | \( 1 - 12.5iT - 73T^{2} \) |
| 79 | \( 1 + 14.0iT - 79T^{2} \) |
| 83 | \( 1 + (10.9 + 10.9i)T + 83iT^{2} \) |
| 89 | \( 1 + 0.412T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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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
−8.792387606858923325540181530486, −7.989078845322859943866908593773, −7.51474288585950380822639653432, −6.66710134135122853552382353576, −5.36416152676950938137068638047, −4.82060650034784044740253509213, −4.27425082961107888534931230514, −2.91302280461639818312165637377, −1.87723647096556235121743694403, −0.61018952234796404060463052014,
1.29050067723951266510153439638, 2.32878331314280958794470295795, 3.62990224156568803687136199173, 4.13379460910137007616243850809, 5.45880333944757938962819115085, 5.76143139687334168249034041981, 7.07557635536242022966855536286, 7.80754315401646725900367728478, 8.155969693258835455806741158791, 9.006039080724196545779790453976