| L(s) = 1 | − 4.96e4i·2-s − 1.55e7i·3-s + 6.12e9·4-s − 7.71e11·6-s + 1.22e14i·7-s − 7.30e14i·8-s + 5.31e15·9-s + 2.15e17·11-s − 9.50e16i·12-s − 1.07e18i·13-s + 6.06e18·14-s + 1.62e19·16-s − 2.54e20i·17-s − 2.64e20i·18-s + 1.22e21·19-s + ⋯ |
| L(s) = 1 | − 0.536i·2-s − 0.208i·3-s + 0.712·4-s − 0.111·6-s + 1.38i·7-s − 0.918i·8-s + 0.956·9-s + 1.41·11-s − 0.148i·12-s − 0.449i·13-s + 0.744·14-s + 0.220·16-s − 1.26i·17-s − 0.512i·18-s + 0.970·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(17)\) |
\(\approx\) |
\(3.949408400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.949408400\) |
| \(L(\frac{35}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 4.96e4iT - 8.58e9T^{2} \) |
| 3 | \( 1 + 1.55e7iT - 5.55e15T^{2} \) |
| 7 | \( 1 - 1.22e14iT - 7.73e27T^{2} \) |
| 11 | \( 1 - 2.15e17T + 2.32e34T^{2} \) |
| 13 | \( 1 + 1.07e18iT - 5.75e36T^{2} \) |
| 17 | \( 1 + 2.54e20iT - 4.02e40T^{2} \) |
| 19 | \( 1 - 1.22e21T + 1.58e42T^{2} \) |
| 23 | \( 1 + 5.11e21iT - 8.65e44T^{2} \) |
| 29 | \( 1 - 1.64e23T + 1.81e48T^{2} \) |
| 31 | \( 1 + 6.75e24T + 1.64e49T^{2} \) |
| 37 | \( 1 - 7.46e25iT - 5.63e51T^{2} \) |
| 41 | \( 1 - 4.96e26T + 1.66e53T^{2} \) |
| 43 | \( 1 + 1.99e26iT - 8.02e53T^{2} \) |
| 47 | \( 1 + 2.16e27iT - 1.51e55T^{2} \) |
| 53 | \( 1 + 3.60e28iT - 7.96e56T^{2} \) |
| 59 | \( 1 - 1.87e29T + 2.74e58T^{2} \) |
| 61 | \( 1 - 4.18e27T + 8.23e58T^{2} \) |
| 67 | \( 1 + 4.85e29iT - 1.82e60T^{2} \) |
| 71 | \( 1 + 3.42e30T + 1.23e61T^{2} \) |
| 73 | \( 1 - 7.01e30iT - 3.08e61T^{2} \) |
| 79 | \( 1 - 2.95e30T + 4.18e62T^{2} \) |
| 83 | \( 1 + 1.23e31iT - 2.13e63T^{2} \) |
| 89 | \( 1 + 7.05e31T + 2.13e64T^{2} \) |
| 97 | \( 1 - 7.71e32iT - 3.65e65T^{2} \) |
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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
−11.44875385333003010890027909293, −9.897820951206062273642017185823, −9.061841415303191155910142204653, −7.42643418739594355350047933479, −6.52073576851840968158247973157, −5.31248398465302009492686864020, −3.74067515513565405365706615506, −2.66075507469925669054284849425, −1.70607888546296212809527741994, −0.804767480488882017166037510230,
1.07153670297011045903868857601, 1.74554174370435379989627101769, 3.60001395140453920284843399498, 4.30592944987151372285920308095, 5.94057826848992868277443194776, 7.01283143231191852436439322892, 7.55860738266818492370762314962, 9.205575240951467774760054242095, 10.42971163212276633867932130912, 11.31176630123104991851672265781
