| L(s) = 1 | + (−1.51 + 1.30i)2-s + (−2.22 − 2.01i)3-s + (0.587 − 3.95i)4-s + (−1.88 − 7.04i)5-s + (5.99 + 0.153i)6-s + (−8.89 + 5.13i)7-s + (4.27 + 6.75i)8-s + (0.864 + 8.95i)9-s + (12.0 + 8.20i)10-s + (8.94 + 2.39i)11-s + (−9.28 + 7.60i)12-s + (4.68 + 17.4i)13-s + (6.76 − 19.4i)14-s + (−10.0 + 19.4i)15-s + (−15.3 − 4.64i)16-s − 16.2i·17-s + ⋯ |
| L(s) = 1 | + (−0.757 + 0.653i)2-s + (−0.740 − 0.672i)3-s + (0.146 − 0.989i)4-s + (−0.377 − 1.40i)5-s + (0.999 + 0.0255i)6-s + (−1.27 + 0.733i)7-s + (0.534 + 0.844i)8-s + (0.0960 + 0.995i)9-s + (1.20 + 0.820i)10-s + (0.813 + 0.217i)11-s + (−0.773 + 0.633i)12-s + (0.360 + 1.34i)13-s + (0.483 − 1.38i)14-s + (−0.667 + 1.29i)15-s + (−0.956 − 0.290i)16-s − 0.953i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.589 - 0.807i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.589 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0948323 + 0.186638i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0948323 + 0.186638i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.51 - 1.30i)T \) |
| 3 | \( 1 + (2.22 + 2.01i)T \) |
| good | 5 | \( 1 + (1.88 + 7.04i)T + (-21.6 + 12.5i)T^{2} \) |
| 7 | \( 1 + (8.89 - 5.13i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-8.94 - 2.39i)T + (104. + 60.5i)T^{2} \) |
| 13 | \( 1 + (-4.68 - 17.4i)T + (-146. + 84.5i)T^{2} \) |
| 17 | \( 1 + 16.2iT - 289T^{2} \) |
| 19 | \( 1 + (10.1 - 10.1i)T - 361iT^{2} \) |
| 23 | \( 1 + (15.7 - 27.3i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-5.89 + 21.9i)T + (-728. - 420.5i)T^{2} \) |
| 31 | \( 1 + (4.70 - 8.15i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (9.86 + 9.86i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + (3.91 - 6.77i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (17.7 - 66.3i)T + (-1.60e3 - 924.5i)T^{2} \) |
| 47 | \( 1 + (47.8 - 27.6i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (62.9 - 62.9i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (8.22 + 30.7i)T + (-3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-33.4 - 8.95i)T + (3.22e3 + 1.86e3i)T^{2} \) |
| 67 | \( 1 + (11.9 + 44.7i)T + (-3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + 51.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 12.1iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-5.26 - 9.12i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-11.6 + 43.5i)T + (-5.96e3 - 3.44e3i)T^{2} \) |
| 89 | \( 1 + 48.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-8.69 - 15.0i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−13.07771031017721241024406843577, −12.07391976269225908022112150096, −11.47926808618630511555695146373, −9.654881738320699155343064494353, −9.119143851542495052242254817016, −7.962528885370879937219165228312, −6.66353277839947538285215010176, −5.91967638847339346807028591649, −4.59533959979151239229519695924, −1.53511755434253786204408338233,
0.19620734941261894681906906345, 3.21921972462431652482448347289, 3.87245967343712659636385406007, 6.35116648528032214982831158186, 6.93692027010142261878565432880, 8.520512027827106587398635962497, 9.972618984730577201532399358987, 10.46265837658852258461766010702, 11.05369353059459793358101876496, 12.23171320683053177409989778314
