L(s) = 1 | + (−0.991 − 0.130i)2-s + (0.965 + 0.258i)4-s + (−1.09 + 1.25i)5-s + (−0.923 − 0.382i)8-s + (0.608 + 0.793i)9-s + (1.25 − 1.09i)10-s + (1 − i)13-s + (0.866 + 0.5i)16-s + (0.793 + 0.608i)17-s + (−0.499 − 0.866i)18-s + (−1.38 + 0.923i)20-s + (−0.230 − 1.75i)25-s + (−1.12 + 0.860i)26-s + (1.63 + 0.324i)29-s + (−0.793 − 0.608i)32-s + ⋯ |
L(s) = 1 | + (−0.991 − 0.130i)2-s + (0.965 + 0.258i)4-s + (−1.09 + 1.25i)5-s + (−0.923 − 0.382i)8-s + (0.608 + 0.793i)9-s + (1.25 − 1.09i)10-s + (1 − i)13-s + (0.866 + 0.5i)16-s + (0.793 + 0.608i)17-s + (−0.499 − 0.866i)18-s + (−1.38 + 0.923i)20-s + (−0.230 − 1.75i)25-s + (−1.12 + 0.860i)26-s + (1.63 + 0.324i)29-s + (−0.793 − 0.608i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7362905831\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7362905831\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.991 + 0.130i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.793 - 0.608i)T \) |
good | 3 | \( 1 + (-0.608 - 0.793i)T^{2} \) |
| 5 | \( 1 + (1.09 - 1.25i)T + (-0.130 - 0.991i)T^{2} \) |
| 11 | \( 1 + (0.991 + 0.130i)T^{2} \) |
| 13 | \( 1 + (-1 + i)T - iT^{2} \) |
| 19 | \( 1 + (0.965 + 0.258i)T^{2} \) |
| 23 | \( 1 + (0.608 - 0.793i)T^{2} \) |
| 29 | \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + (0.608 + 0.793i)T^{2} \) |
| 37 | \( 1 + (-1.10 + 0.0726i)T + (0.991 - 0.130i)T^{2} \) |
| 41 | \( 1 + (1.92 - 0.382i)T + (0.923 - 0.382i)T^{2} \) |
| 43 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (-1.12 + 1.46i)T + (-0.258 - 0.965i)T^{2} \) |
| 59 | \( 1 + (-0.965 + 0.258i)T^{2} \) |
| 61 | \( 1 + (0.369 + 0.125i)T + (0.793 + 0.608i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 73 | \( 1 + (-0.357 - 1.05i)T + (-0.793 + 0.608i)T^{2} \) |
| 79 | \( 1 + (-0.608 + 0.793i)T^{2} \) |
| 83 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 97 | \( 1 + (0.216 - 1.08i)T + (-0.923 - 0.382i)T^{2} \) |
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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
−8.670887506489250684401579640540, −8.008294596894279234747075171165, −7.79467216575333618187438625636, −6.82936672075167477825965292241, −6.38237905686025769030136248772, −5.25018032819728443864873653953, −3.91347892450026708685576745645, −3.30778366561517825425808390160, −2.49450625595718552119242791782, −1.15359923355206541403725485313,
0.78485564838477890439651193231, 1.51458560716832772343097917130, 3.10134038671421883305311767163, 4.02095423389818680471157437974, 4.73603477297703203900006386206, 5.82362832064963707846592908577, 6.66601933222541661118361000784, 7.32439364993219009441121442675, 8.133899391967013502234074038055, 8.646280414495032928229046279336