L(s) = 1 | + 25.1i·2-s − 638.·3-s + 1.41e3·4-s − 9.28e3i·5-s − 1.60e4i·6-s + 2.52e4i·7-s + 8.71e4i·8-s + 2.30e5·9-s + 2.33e5·10-s + 8.09e5i·11-s − 9.03e5·12-s + (7.38e5 − 1.11e6i)13-s − 6.34e5·14-s + 5.93e6i·15-s + 7.07e5·16-s + 1.09e7·17-s + ⋯ |
L(s) = 1 | + 0.555i·2-s − 1.51·3-s + 0.691·4-s − 1.32i·5-s − 0.843i·6-s + 0.567i·7-s + 0.939i·8-s + 1.30·9-s + 0.738·10-s + 1.51i·11-s − 1.04·12-s + (0.551 − 0.833i)13-s − 0.315·14-s + 2.01i·15-s + 0.168·16-s + 1.87·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.551 - 0.833i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.551 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(1.12454 + 0.604312i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12454 + 0.604312i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (-7.38e5 + 1.11e6i)T \) |
good | 2 | \( 1 - 25.1iT - 2.04e3T^{2} \) |
| 3 | \( 1 + 638.T + 1.77e5T^{2} \) |
| 5 | \( 1 + 9.28e3iT - 4.88e7T^{2} \) |
| 7 | \( 1 - 2.52e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 - 8.09e5iT - 2.85e11T^{2} \) |
| 17 | \( 1 - 1.09e7T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.08e7iT - 1.16e14T^{2} \) |
| 23 | \( 1 + 5.65e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 7.64e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.22e8iT - 2.54e16T^{2} \) |
| 37 | \( 1 - 5.45e7iT - 1.77e17T^{2} \) |
| 41 | \( 1 - 9.95e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 + 3.96e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 7.01e8iT - 2.47e18T^{2} \) |
| 53 | \( 1 - 6.66e8T + 9.26e18T^{2} \) |
| 59 | \( 1 - 2.42e9iT - 3.01e19T^{2} \) |
| 61 | \( 1 + 8.91e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 6.44e9iT - 1.22e20T^{2} \) |
| 71 | \( 1 - 1.38e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 + 1.48e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 - 2.96e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.25e10iT - 1.28e21T^{2} \) |
| 89 | \( 1 - 9.03e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + 5.24e10iT - 7.15e21T^{2} \) |
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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
−16.95703274166776758637284592430, −16.35549974221503022005704461625, −15.13664322180748992650171293610, −12.46745759306134676383030733385, −11.97640185640651298529589327730, −10.14876109729369904378551672036, −7.87767562116656553467604870153, −5.98702089387829585337725467372, −5.07712789387999113282216955217, −1.26856613898476074537150121438,
0.881357948941809491320370231078, 3.31376408730779791119978360364, 6.00461776131628974997139026487, 7.02881726781072038277187925974, 10.43850869574018190031743747656, 11.03995666380438300238820070942, 11.97415034434512948630544640310, 14.01824055831178442865610724904, 15.95028927555336104768387637728, 16.85707809803685061168196369658