| L(s) = 1 | + (−1.23 − 0.186i)2-s + (−2.40 + 1.63i)3-s + (−0.419 − 0.129i)4-s + (0.186 + 2.49i)5-s + (3.27 − 1.57i)6-s + (−2.47 − 0.946i)7-s + (2.74 + 1.32i)8-s + (1.99 − 5.08i)9-s + (0.233 − 3.11i)10-s + (0.723 + 1.84i)11-s + (1.22 − 0.376i)12-s + (−1.63 + 2.05i)13-s + (2.87 + 1.62i)14-s + (−4.52 − 5.67i)15-s + (−2.41 − 1.64i)16-s + (1.83 + 1.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.873 − 0.131i)2-s + (−1.38 + 0.946i)3-s + (−0.209 − 0.0647i)4-s + (0.0834 + 1.11i)5-s + (1.33 − 0.643i)6-s + (−0.933 − 0.357i)7-s + (0.970 + 0.467i)8-s + (0.665 − 1.69i)9-s + (0.0737 − 0.983i)10-s + (0.218 + 0.555i)11-s + (0.352 − 0.108i)12-s + (−0.454 + 0.570i)13-s + (0.768 + 0.435i)14-s + (−1.16 − 1.46i)15-s + (−0.604 − 0.412i)16-s + (0.444 + 0.412i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.105719 + 0.230555i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.105719 + 0.230555i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.47 + 0.946i)T \) |
| good | 2 | \( 1 + (1.23 + 0.186i)T + (1.91 + 0.589i)T^{2} \) |
| 3 | \( 1 + (2.40 - 1.63i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.186 - 2.49i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (-0.723 - 1.84i)T + (-8.06 + 7.48i)T^{2} \) |
| 13 | \( 1 + (1.63 - 2.05i)T + (-2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (-1.83 - 1.69i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.16 - 2.02i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.68 - 4.34i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.385 - 1.68i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-2.50 + 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.77 + 1.16i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (3.16 + 1.52i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-3.27 + 1.57i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-4.48 - 0.676i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (-3.25 - 1.00i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (0.582 - 7.77i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (12.7 - 3.93i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-4.31 + 7.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.97 - 8.66i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (5.52 - 0.832i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (-8.63 - 14.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.86 - 3.58i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-4.93 + 12.5i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 4.68T + 97T^{2} \) |
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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
−16.47065205307469690631688832636, −15.23407282173921611354492008100, −13.94687661432974552053830760536, −12.14663617613293415994224325067, −10.93130187576842436651698198028, −10.07913984042211804937076782036, −9.576679387366614928357082000626, −7.24882513740871785670853052316, −5.89523487198330994087416932817, −4.11902978839987215288367660293,
0.64563580618313198116007393821, 5.03012855541802805730264614575, 6.37244184108254183030213854486, 7.82347599183033926182765243225, 9.128505371335406867280048255396, 10.37569611584630370153927899709, 11.99626917574324278011937922489, 12.70519595887676120757290759084, 13.57505800347789780768881844233, 16.08589917788992987329470360001
