| L(s) = 1 | + (1.03 + 0.866i)2-s + (−2.70 + 0.984i)3-s + (−0.0320 − 0.181i)4-s + (0.152 − 0.866i)5-s + (−3.64 − 1.32i)6-s + (−0.173 − 0.300i)7-s + (1.47 − 2.54i)8-s + (4.05 − 3.40i)9-s + (0.907 − 0.761i)10-s + (1.11 − 1.92i)11-s + (0.266 + 0.460i)12-s + (2.41 + 0.880i)13-s + (0.0812 − 0.460i)14-s + (0.439 + 2.49i)15-s + (3.37 − 1.22i)16-s + (0.358 + 0.300i)17-s + ⋯ |
| L(s) = 1 | + (0.729 + 0.612i)2-s + (−1.56 + 0.568i)3-s + (−0.0160 − 0.0909i)4-s + (0.0682 − 0.387i)5-s + (−1.48 − 0.541i)6-s + (−0.0656 − 0.113i)7-s + (0.520 − 0.901i)8-s + (1.35 − 1.13i)9-s + (0.287 − 0.240i)10-s + (0.335 − 0.581i)11-s + (0.0768 + 0.133i)12-s + (0.670 + 0.244i)13-s + (0.0217 − 0.123i)14-s + (0.113 + 0.643i)15-s + (0.844 − 0.307i)16-s + (0.0869 + 0.0729i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19096 - 0.0906990i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19096 - 0.0906990i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 \) |
| good | 2 | \( 1 + (-1.03 - 0.866i)T + (0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (2.70 - 0.984i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-0.152 + 0.866i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.173 + 0.300i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.11 + 1.92i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.41 - 0.880i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.358 - 0.300i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.467 + 2.65i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.26 + 4.42i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (3.55 + 6.15i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.94T + 37T^{2} \) |
| 41 | \( 1 + (-2.32 + 0.846i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.677 + 3.84i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (5.58 - 4.68i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.492 - 2.79i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-4.83 - 4.05i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.58 - 8.98i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (5.87 - 4.93i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.61 + 9.16i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (1.30 - 0.475i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (11.1 - 4.05i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.41 - 12.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (9.67 + 3.51i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (7.24 + 6.07i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.40333010510959500134181174399, −10.61498361768112918319051969727, −9.858421461550462655229398527195, −8.703922277043573595319411485184, −7.07946453912274127228933629482, −6.18462197069025586537541959074, −5.64909453201214905204897591622, −4.67736220734777978698915476600, −3.89220519894276574698890770333, −0.894373261433796014823758858650,
1.57282376578492670554239991656, 3.24391198022789388023051615787, 4.63702876406619095600154636592, 5.44968760164206810851468945528, 6.53190391954744928841310496462, 7.31098419864690644920146273633, 8.604546488474477853786312653878, 10.19413661858309263943036197962, 10.94475575174505044699232046005, 11.54456998574083078151198736512
