| L(s) = 1 | − i·2-s + 2.77i·3-s − 4-s + 2.77·6-s + 4.69i·7-s + i·8-s − 4.71·9-s + 6.40·11-s − 2.77i·12-s + 1.06i·13-s + 4.69·14-s + 16-s − 1.91i·17-s + 4.71i·18-s + 19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 1.60i·3-s − 0.5·4-s + 1.13·6-s + 1.77i·7-s + 0.353i·8-s − 1.57·9-s + 1.93·11-s − 0.801i·12-s + 0.295i·13-s + 1.25·14-s + 0.250·16-s − 0.465i·17-s + 1.11i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.765086 + 1.23793i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.765086 + 1.23793i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.77iT - 3T^{2} \) |
| 7 | \( 1 - 4.69iT - 7T^{2} \) |
| 11 | \( 1 - 6.40T + 11T^{2} \) |
| 13 | \( 1 - 1.06iT - 13T^{2} \) |
| 17 | \( 1 + 1.91iT - 17T^{2} \) |
| 23 | \( 1 - 1.79iT - 23T^{2} \) |
| 29 | \( 1 + 2.93T + 29T^{2} \) |
| 31 | \( 1 + 5.55T + 31T^{2} \) |
| 37 | \( 1 - 11.4iT - 37T^{2} \) |
| 41 | \( 1 + 1.14T + 41T^{2} \) |
| 43 | \( 1 + 3.55iT - 43T^{2} \) |
| 47 | \( 1 + 10.8iT - 47T^{2} \) |
| 53 | \( 1 + 8.69iT - 53T^{2} \) |
| 59 | \( 1 - 5.63T + 59T^{2} \) |
| 61 | \( 1 + 3.39T + 61T^{2} \) |
| 67 | \( 1 - 8.82iT - 67T^{2} \) |
| 71 | \( 1 + 1.42T + 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 1.96T + 79T^{2} \) |
| 83 | \( 1 - 16.2iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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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
−10.11212751921513873503012246994, −9.407280237894318795812593156932, −9.077135891723693581284375331055, −8.437913938842757904314995514033, −6.68045117364986196385685634605, −5.60074215728279451334178434477, −4.95351172314408700806764044554, −3.89379226034965580745316953999, −3.21113583497153760733090732349, −1.89896643138352464006313171727,
0.73686059420716282848172966438, 1.62392006404824758751436284376, 3.56189227257520263929000837317, 4.36182395210620741668243346284, 5.94803329328573570962246717951, 6.52403714123786281146950932916, 7.38453772919985928934438155818, 7.54372874411936536480041839811, 8.710156232455901355877032617027, 9.523004732371716128594580202863
