| L(s) = 1 | + (−0.0769 + 0.104i)2-s + (−1.66 + 0.314i)3-s + (0.584 + 1.89i)4-s + (−1.50 − 1.64i)5-s + (0.0950 − 0.197i)6-s + (2.27 + 1.34i)7-s + (−0.486 − 0.170i)8-s + (−0.131 + 0.0517i)9-s + (0.288 − 0.0303i)10-s + (−1.23 + 3.15i)11-s + (−1.56 − 2.96i)12-s + (−6.14 − 0.692i)13-s + (−0.315 + 0.134i)14-s + (3.02 + 2.26i)15-s + (−3.22 + 2.19i)16-s + (−0.450 − 0.0168i)17-s + ⋯ |
| L(s) = 1 | + (−0.0543 + 0.0736i)2-s + (−0.959 + 0.181i)3-s + (0.292 + 0.947i)4-s + (−0.674 − 0.737i)5-s + (0.0387 − 0.0805i)6-s + (0.861 + 0.507i)7-s + (−0.172 − 0.0602i)8-s + (−0.0439 + 0.0172i)9-s + (0.0910 − 0.00961i)10-s + (−0.372 + 0.950i)11-s + (−0.452 − 0.855i)12-s + (−1.70 − 0.191i)13-s + (−0.0842 + 0.0358i)14-s + (0.781 + 0.585i)15-s + (−0.805 + 0.549i)16-s + (−0.109 − 0.00408i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.825 - 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.150483 + 0.486279i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.150483 + 0.486279i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.50 + 1.64i)T \) |
| 7 | \( 1 + (-2.27 - 1.34i)T \) |
| good | 2 | \( 1 + (0.0769 - 0.104i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (1.66 - 0.314i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (1.23 - 3.15i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (6.14 + 0.692i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (0.450 + 0.0168i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (3.38 - 5.86i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.143 + 3.82i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (-3.87 + 0.883i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-4.62 + 2.67i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.20 - 0.637i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (-5.04 - 10.4i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-2.89 - 8.27i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (4.33 + 3.19i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (0.663 + 0.350i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (0.851 + 11.3i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (0.152 - 0.494i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (0.342 + 0.0916i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.67 + 7.34i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-5.86 + 4.32i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (-3.12 - 1.80i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.61 - 14.3i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (-1.14 - 2.92i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-7.85 - 7.85i)T + 97iT^{2} \) |
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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
−12.24705382419367008716706400146, −11.82614461017182725113573629415, −10.84076487945981503817511254575, −9.632039108801118986926149275902, −8.101302420144672118506629922323, −7.932672472760577762970604628435, −6.44927240415275897494447143772, −4.96638100910726892007359851128, −4.46839132028482846092379438326, −2.43296062557752847705474106393,
0.43937158932039992642674523074, 2.59562867832540792362280618640, 4.60766521261094333541853112998, 5.50979580318587182846714925409, 6.71057279999473653801263877141, 7.38977037672313899123291431962, 8.797083932051176537241408279686, 10.31157516970134815932213959126, 10.85735155900737532616855854613, 11.49953739043872111114774962278
