| L(s) = 1 | − 3i·3-s − 12.5·5-s + (−18.2 + 2.88i)7-s − 9·9-s + 55.4·11-s − 92.1·13-s + 37.5i·15-s − 118. i·17-s − 155. i·19-s + (8.66 + 54.8i)21-s − 125. i·23-s + 31.6·25-s + 27i·27-s + 131. i·29-s + 66.0·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.11·5-s + (−0.987 + 0.155i)7-s − 0.333·9-s + 1.51·11-s − 1.96·13-s + 0.646i·15-s − 1.68i·17-s − 1.87i·19-s + (0.0900 + 0.570i)21-s − 1.13i·23-s + 0.253·25-s + 0.192i·27-s + 0.842i·29-s + 0.382·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.05001367758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05001367758\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| 7 | \( 1 + (18.2 - 2.88i)T \) |
| good | 5 | \( 1 + 12.5T + 125T^{2} \) |
| 11 | \( 1 - 55.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 92.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 118. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 155. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 125. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 131. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 66.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 147. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 20.3iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 355.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 79.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 463. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 580. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 587.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 496.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 232. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 551. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 437. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 191. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 93.7iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 758. iT - 9.12e5T^{2} \) |
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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
−8.812446334500510098673161258550, −7.46351455249182967423421790234, −7.04620173463821134972894310958, −6.54725827195284842851704791912, −5.07491654449171739316664475407, −4.37359529008094330596106879210, −3.14586673162647661193293349038, −2.46480593349087796199948536958, −0.65364338877483886148258078153, −0.01833822487104721315956187388,
1.65021463743472067469602616073, 3.25479386072848475014671183045, 3.85253451138607159083824400327, 4.47639687504918040897914406141, 5.82444683164711954467236456273, 6.53883225512146495750282115752, 7.56811398824907250939018487760, 8.137395000487861866324915709905, 9.238407588101269380762090415060, 9.868612350347041118709529311154
