L(s) = 1 | + (0.342 − 1.97i)2-s + (−3.76 − 1.34i)4-s + 11.5i·7-s + (−3.94 + 6.95i)8-s − 9.97i·11-s + 14.1·13-s + (22.6 + 3.94i)14-s + (12.3 + 10.1i)16-s − 30.5·17-s − 12.2i·19-s + (−19.6 − 3.41i)22-s − 15.7i·23-s + (4.83 − 27.8i)26-s + (15.5 − 43.3i)28-s − 18.8·29-s + ⋯ |
L(s) = 1 | + (0.171 − 0.985i)2-s + (−0.941 − 0.337i)4-s + 1.64i·7-s + (−0.493 + 0.869i)8-s − 0.906i·11-s + 1.08·13-s + (1.62 + 0.281i)14-s + (0.772 + 0.635i)16-s − 1.79·17-s − 0.643i·19-s + (−0.893 − 0.155i)22-s − 0.685i·23-s + (0.185 − 1.07i)26-s + (0.554 − 1.54i)28-s − 0.651·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.337i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5096332057\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5096332057\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.342 + 1.97i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 11.5iT - 49T^{2} \) |
| 11 | \( 1 + 9.97iT - 121T^{2} \) |
| 13 | \( 1 - 14.1T + 169T^{2} \) |
| 17 | \( 1 + 30.5T + 289T^{2} \) |
| 19 | \( 1 + 12.2iT - 361T^{2} \) |
| 23 | \( 1 + 15.7iT - 529T^{2} \) |
| 29 | \( 1 + 18.8T + 841T^{2} \) |
| 31 | \( 1 + 35.2iT - 961T^{2} \) |
| 37 | \( 1 + 50.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 28.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 24.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 55.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.46T + 2.80e3T^{2} \) |
| 59 | \( 1 + 64.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 30.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 66.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 11.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 2.24T + 5.32e3T^{2} \) |
| 79 | \( 1 - 78.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 146. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 87.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 126.T + 9.40e3T^{2} \) |
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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
−9.288075418976129964885104173646, −8.755499610098345630104581105096, −8.355124326845089201500201729335, −6.54145636586516323686972833486, −5.80794748157160730944269719457, −4.94896461624498428355044434718, −3.79850524364111506364706499644, −2.73571454388158760664952669255, −1.89262971932763988384360120031, −0.15735355407370441370294699395,
1.46222997343255004720263186432, 3.55818284514921489450721881029, 4.19566121581534425550944968195, 5.06259342631449495301940270381, 6.34077324762085209387716735398, 6.97375856851302689175448479257, 7.60788111282477112822746581341, 8.576129629842552517134294513811, 9.367041462947503044330979732303, 10.37213623758758044516466219775