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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 + 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.367096610\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.367096610\) |
\(L(\frac{3}{2})\) |
|
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
−8.786982086264686532453499871062, −8.245851593046610928694322825348, −7.41432446542187602552107607381, −6.58601129686347427921469930858, −5.84337694437962561444232545110, −4.87376427963857787662341277750, −4.28778706879284010802584125868, −2.89341074102006343304222288773, −2.17142502335563965851268554063, −0.868148695464757876743512507751,
1.47856058006570267982574110386, 1.79991504647313567368507973850, 3.36962642407594227221395934991, 4.51413636213396041616342853673, 4.94431317693666528686136921082, 5.89893794769173022888934307189, 6.69230022830672875188206755873, 7.77363897067387627437262474249, 8.215057169515360925484146819709, 9.132842247867458027835740283345