| L(s) = 1 | + (1.61 − 1.17i)2-s + (0.292 + 0.901i)3-s + (1.23 − 3.80i)4-s + (4.04 + 2.93i)5-s + (1.53 + 1.11i)6-s + (2.73 − 8.41i)7-s + (−2.47 − 7.60i)8-s + (21.1 − 15.3i)9-s + 10·10-s + (36.3 + 3.21i)11-s + 3.79·12-s + (2.56 − 1.86i)13-s + (−5.46 − 16.8i)14-s + (−1.46 + 4.50i)15-s + (−12.9 − 9.40i)16-s + (16.7 + 12.1i)17-s + ⋯ |
| L(s) = 1 | + (0.572 − 0.415i)2-s + (0.0563 + 0.173i)3-s + (0.154 − 0.475i)4-s + (0.361 + 0.262i)5-s + (0.104 + 0.0757i)6-s + (0.147 − 0.454i)7-s + (−0.109 − 0.336i)8-s + (0.782 − 0.568i)9-s + 0.316·10-s + (0.996 + 0.0881i)11-s + 0.0911·12-s + (0.0546 − 0.0397i)13-s + (−0.104 − 0.321i)14-s + (−0.0252 + 0.0775i)15-s + (−0.202 − 0.146i)16-s + (0.239 + 0.173i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.30080 - 0.892840i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.30080 - 0.892840i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.61 + 1.17i)T \) |
| 5 | \( 1 + (-4.04 - 2.93i)T \) |
| 11 | \( 1 + (-36.3 - 3.21i)T \) |
| good | 3 | \( 1 + (-0.292 - 0.901i)T + (-21.8 + 15.8i)T^{2} \) |
| 7 | \( 1 + (-2.73 + 8.41i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-2.56 + 1.86i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-16.7 - 12.1i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (14.3 + 44.2i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 43.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + (13.0 - 40.0i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (177. - 129. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (18.4 - 56.8i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-22.2 - 68.5i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 368.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-83.7 - 257. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (460. - 334. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-41.7 + 128. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (61.3 + 44.5i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 113.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (458. + 333. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (215. - 661. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-590. + 428. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (924. + 671. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-186. + 135. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−13.03492399378651129323639061788, −12.11482171655512763353171240966, −10.97955991924758293313335293073, −10.01486204400581518037910477213, −9.050860902788494435512807669128, −7.22015598294542195166866029520, −6.21111910319215405663270185075, −4.59554029092388674146781058612, −3.44839032844192127786490174052, −1.46364707852849424203940481894,
1.89169770517639109155226621848, 3.91720347718470053841834797362, 5.26873521580286128125988650321, 6.44729556227921941692255667658, 7.66270566153220231078969519544, 8.857012785562844139846156790905, 10.06060843938190990550091767720, 11.50989739757048926926578388379, 12.45645143323966976279389075854, 13.36208250479887831870688617184
