L(s) = 1 | + (1.36 + 0.366i)2-s + (−1.70 + 0.554i)3-s + (1.73 + 1.00i)4-s + (−2.39 − 1.74i)5-s + (−2.53 + 0.132i)6-s + (0.815 − 2.51i)7-s + (1.99 + 2.00i)8-s + (0.174 − 0.126i)9-s + (−2.63 − 3.26i)10-s + (−1.40 + 3.00i)11-s + (−3.50 − 0.747i)12-s + (1.39 + 1.92i)13-s + (2.03 − 3.13i)14-s + (5.05 + 1.64i)15-s + (1.99 + 3.46i)16-s + (0.468 − 0.644i)17-s + ⋯ |
L(s) = 1 | + (0.965 + 0.259i)2-s + (−0.984 + 0.319i)3-s + (0.865 + 0.500i)4-s + (−1.07 − 0.779i)5-s + (−1.03 + 0.0539i)6-s + (0.308 − 0.948i)7-s + (0.706 + 0.707i)8-s + (0.0580 − 0.0421i)9-s + (−0.834 − 1.03i)10-s + (−0.425 + 0.905i)11-s + (−1.01 − 0.215i)12-s + (0.388 + 0.534i)13-s + (0.543 − 0.836i)14-s + (1.30 + 0.424i)15-s + (0.499 + 0.866i)16-s + (0.113 − 0.156i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.907889 + 0.166659i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.907889 + 0.166659i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.36 - 0.366i)T \) |
| 11 | \( 1 + (1.40 - 3.00i)T \) |
good | 3 | \( 1 + (1.70 - 0.554i)T + (2.42 - 1.76i)T^{2} \) |
| 5 | \( 1 + (2.39 + 1.74i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.815 + 2.51i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-1.39 - 1.92i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.468 + 0.644i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.624 + 1.92i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 2.61iT - 23T^{2} \) |
| 29 | \( 1 + (1.08 + 0.351i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.51 - 4.84i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.69 + 8.30i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (9.28 - 3.01i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 6.14T + 43T^{2} \) |
| 47 | \( 1 + (3.31 - 1.07i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-6.63 + 4.82i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.712 + 0.231i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (3.71 - 5.11i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 5.40iT - 67T^{2} \) |
| 71 | \( 1 + (-2.56 + 3.53i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.845 - 0.274i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-2.17 + 1.58i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-12.1 - 8.85i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 4.25T + 89T^{2} \) |
| 97 | \( 1 + (5.92 - 4.30i)T + (29.9 - 92.2i)T^{2} \) |
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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
−16.16048181475930077477007359129, −15.02851644067673355183612157574, −13.62129907683157087299827304335, −12.37441742274047673044304721838, −11.54528231449592106268446951384, −10.55316959368569257632896860251, −8.166427226356224603415539164854, −6.88065201305950286686064102684, −5.02909577387759671992870111748, −4.21844425640318850237605637302,
3.29789081753004457615719220340, 5.38726880851750830540119245571, 6.44039295670537873745539656767, 8.049616327185247093804039339896, 10.59674308782341090692694324536, 11.54698823036479209096413876670, 11.99850040859835058959925269508, 13.40522098115577928413217851045, 14.94102641436383813536834837757, 15.52065305105167513226197453227