| L(s) = 1 | + (0.247 − 0.0268i)2-s + (0.647 + 0.762i)3-s + (−1.89 + 0.416i)4-s + (−2.60 + 1.97i)5-s + (0.180 + 0.170i)6-s + (0.505 + 1.26i)7-s + (−0.927 + 0.312i)8-s + (−0.161 + 0.986i)9-s + (−0.589 + 0.558i)10-s + (0.272 + 0.126i)11-s + (−1.54 − 1.17i)12-s + (0.501 + 3.05i)13-s + (0.158 + 0.299i)14-s + (−3.19 − 0.702i)15-s + (3.29 − 1.52i)16-s + (−0.182 + 0.457i)17-s + ⋯ |
| L(s) = 1 | + (0.174 − 0.0189i)2-s + (0.373 + 0.440i)3-s + (−0.946 + 0.208i)4-s + (−1.16 + 0.884i)5-s + (0.0736 + 0.0697i)6-s + (0.190 + 0.479i)7-s + (−0.327 + 0.110i)8-s + (−0.0539 + 0.328i)9-s + (−0.186 + 0.176i)10-s + (0.0822 + 0.0380i)11-s + (−0.445 − 0.338i)12-s + (0.138 + 0.847i)13-s + (0.0424 + 0.0800i)14-s + (−0.824 − 0.181i)15-s + (0.824 − 0.381i)16-s + (−0.0442 + 0.111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.489924 + 0.726186i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.489924 + 0.726186i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.647 - 0.762i)T \) |
| 59 | \( 1 + (-4.65 + 6.11i)T \) |
| good | 2 | \( 1 + (-0.247 + 0.0268i)T + (1.95 - 0.429i)T^{2} \) |
| 5 | \( 1 + (2.60 - 1.97i)T + (1.33 - 4.81i)T^{2} \) |
| 7 | \( 1 + (-0.505 - 1.26i)T + (-5.08 + 4.81i)T^{2} \) |
| 11 | \( 1 + (-0.272 - 0.126i)T + (7.12 + 8.38i)T^{2} \) |
| 13 | \( 1 + (-0.501 - 3.05i)T + (-12.3 + 4.15i)T^{2} \) |
| 17 | \( 1 + (0.182 - 0.457i)T + (-12.3 - 11.6i)T^{2} \) |
| 19 | \( 1 + (-3.88 + 5.73i)T + (-7.03 - 17.6i)T^{2} \) |
| 23 | \( 1 + (-0.233 - 4.31i)T + (-22.8 + 2.48i)T^{2} \) |
| 29 | \( 1 + (4.09 + 0.445i)T + (28.3 + 6.23i)T^{2} \) |
| 31 | \( 1 + (-3.49 - 5.15i)T + (-11.4 + 28.7i)T^{2} \) |
| 37 | \( 1 + (-4.02 - 1.35i)T + (29.4 + 22.3i)T^{2} \) |
| 41 | \( 1 + (0.372 - 6.87i)T + (-40.7 - 4.43i)T^{2} \) |
| 43 | \( 1 + (-1.23 + 0.572i)T + (27.8 - 32.7i)T^{2} \) |
| 47 | \( 1 + (7.70 + 5.86i)T + (12.5 + 45.2i)T^{2} \) |
| 53 | \( 1 + (-1.03 - 0.984i)T + (2.86 + 52.9i)T^{2} \) |
| 61 | \( 1 + (-9.23 + 1.00i)T + (59.5 - 13.1i)T^{2} \) |
| 67 | \( 1 + (12.5 - 4.21i)T + (53.3 - 40.5i)T^{2} \) |
| 71 | \( 1 + (-9.31 - 7.08i)T + (18.9 + 68.4i)T^{2} \) |
| 73 | \( 1 + (3.53 + 6.66i)T + (-40.9 + 60.4i)T^{2} \) |
| 79 | \( 1 + (6.43 - 7.57i)T + (-12.7 - 77.9i)T^{2} \) |
| 83 | \( 1 + (-12.6 - 7.62i)T + (38.8 + 73.3i)T^{2} \) |
| 89 | \( 1 + (6.10 + 0.663i)T + (86.9 + 19.1i)T^{2} \) |
| 97 | \( 1 + (5.00 - 9.43i)T + (-54.4 - 80.2i)T^{2} \) |
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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
−13.15364611980866203572564562137, −11.75844914540104454683264407312, −11.31593638538870875950964892182, −9.851292076574839335236692804693, −8.947839132992054676128181883025, −8.010940496596334718587173118241, −6.93328444943575834223336793019, −5.14795570795958629907983864837, −4.03413114528328801453436089708, −3.05267329845056351450645337489,
0.803819304309675041410896032362, 3.58205049031006103518561186067, 4.50729746022606710404662396455, 5.77881270670157495644628973673, 7.63277329766783823542853886092, 8.165493184410893115860519887638, 9.144006543164728574199677662642, 10.30695122871660817371694896434, 11.71250385581842643562067012438, 12.56903145063794331743271488932
