| L(s) = 1 | + (0.624 + 1.92i)2-s + (3.16 − 2.29i)4-s + (9.66 + 5.61i)5-s + 24.6·7-s + (19.4 + 14.1i)8-s + (−4.76 + 22.1i)10-s + (−21.6 − 66.5i)11-s + (4.46 − 13.7i)13-s + (15.4 + 47.4i)14-s + (−5.38 + 16.5i)16-s + (−9.49 − 6.89i)17-s + (−59.4 − 43.1i)19-s + (43.4 − 4.44i)20-s + (114. − 83.1i)22-s + (38.8 + 119. i)23-s + ⋯ |
| L(s) = 1 | + (0.220 + 0.680i)2-s + (0.395 − 0.287i)4-s + (0.864 + 0.502i)5-s + 1.33·7-s + (0.861 + 0.625i)8-s + (−0.150 + 0.699i)10-s + (−0.592 − 1.82i)11-s + (0.0953 − 0.293i)13-s + (0.294 + 0.906i)14-s + (−0.0842 + 0.259i)16-s + (−0.135 − 0.0983i)17-s + (−0.717 − 0.521i)19-s + (0.486 − 0.0497i)20-s + (1.10 − 0.806i)22-s + (0.352 + 1.08i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.836 - 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.88117 + 0.861009i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.88117 + 0.861009i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-9.66 - 5.61i)T \) |
| good | 2 | \( 1 + (-0.624 - 1.92i)T + (-6.47 + 4.70i)T^{2} \) |
| 7 | \( 1 - 24.6T + 343T^{2} \) |
| 11 | \( 1 + (21.6 + 66.5i)T + (-1.07e3 + 782. i)T^{2} \) |
| 13 | \( 1 + (-4.46 + 13.7i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (9.49 + 6.89i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (59.4 + 43.1i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + (-38.8 - 119. i)T + (-9.84e3 + 7.15e3i)T^{2} \) |
| 29 | \( 1 + (-49.7 + 36.1i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-18.6 - 13.5i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (42.8 - 131. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (105. - 324. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 302.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (65.5 - 47.6i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-29.7 + 21.6i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-57.6 + 177. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-131. - 403. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + (712. + 517. i)T + (9.29e4 + 2.86e5i)T^{2} \) |
| 71 | \( 1 + (316. - 230. i)T + (1.10e5 - 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-141. - 434. i)T + (-3.14e5 + 2.28e5i)T^{2} \) |
| 79 | \( 1 + (101. - 73.7i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-181. - 131. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + (-156. - 482. i)T + (-5.70e5 + 4.14e5i)T^{2} \) |
| 97 | \( 1 + (-1.11e3 + 810. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−11.41783303434606876633314306920, −11.07065022400472565419729097354, −10.17592883072777786775343313236, −8.623120727743079923524936951758, −7.84929629739899832332978751596, −6.62015667861093405969699389883, −5.71099778722230220200008817540, −4.96182308562797884891499938520, −2.88201637028542334868003500615, −1.45986090304123799784423703343,
1.66250855329126529306430583666, 2.27948379933237652309518415471, 4.32680413552303071411492259497, 5.04324100413541401411526042287, 6.67684724005503743161151666945, 7.73769596538121959292820974147, 8.788624956072521379034987534228, 10.16673756144411651655306254588, 10.64104466118332761920337515400, 11.89329566652326399927887107310
