| L(s) = 1 | + (−2.71 + 0.710i)2-s + (−1.73 − 1.47i)3-s + (5.09 − 2.87i)4-s + (2.16 − 2.38i)5-s + (5.74 + 2.76i)6-s + (−0.311 − 0.0815i)7-s + (−7.75 + 7.51i)8-s + (0.347 + 2.14i)9-s + (−4.17 + 8.00i)10-s + (0.579 − 1.30i)11-s + (−13.0 − 2.54i)12-s + (5.35 − 0.690i)13-s + 0.901·14-s + (−7.26 + 0.937i)15-s + (9.61 − 15.8i)16-s + (−1.74 − 0.112i)17-s + ⋯ |
| L(s) = 1 | + (−1.91 + 0.502i)2-s + (−1.00 − 0.851i)3-s + (2.54 − 1.43i)4-s + (0.968 − 1.06i)5-s + (2.34 + 1.12i)6-s + (−0.117 − 0.0308i)7-s + (−2.74 + 2.65i)8-s + (0.115 + 0.715i)9-s + (−1.32 + 2.53i)10-s + (0.174 − 0.394i)11-s + (−3.77 − 0.734i)12-s + (1.48 − 0.191i)13-s + 0.240·14-s + (−1.87 + 0.241i)15-s + (2.40 − 3.96i)16-s + (−0.423 − 0.0272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.340 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.227809 - 0.324889i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.227809 - 0.324889i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (3.06 + 13.6i)T \) |
| good | 2 | \( 1 + (2.71 - 0.710i)T + (1.74 - 0.981i)T^{2} \) |
| 3 | \( 1 + (1.73 + 1.47i)T + (0.478 + 2.96i)T^{2} \) |
| 5 | \( 1 + (-2.16 + 2.38i)T + (-0.480 - 4.97i)T^{2} \) |
| 7 | \( 1 + (0.311 + 0.0815i)T + (6.09 + 3.43i)T^{2} \) |
| 11 | \( 1 + (-0.579 + 1.30i)T + (-7.39 - 8.14i)T^{2} \) |
| 13 | \( 1 + (-5.35 + 0.690i)T + (12.5 - 3.29i)T^{2} \) |
| 17 | \( 1 + (1.74 + 0.112i)T + (16.8 + 2.17i)T^{2} \) |
| 19 | \( 1 + (0.225 + 0.989i)T + (-17.1 + 8.24i)T^{2} \) |
| 23 | \( 1 + (5.87 + 4.38i)T + (6.54 + 22.0i)T^{2} \) |
| 29 | \( 1 + (0.371 + 0.0724i)T + (26.8 + 10.8i)T^{2} \) |
| 31 | \( 1 + (-0.0695 - 2.16i)T + (-30.9 + 1.98i)T^{2} \) |
| 37 | \( 1 + (3.78 + 6.25i)T + (-17.1 + 32.8i)T^{2} \) |
| 41 | \( 1 + (-0.272 - 0.0175i)T + (40.6 + 5.24i)T^{2} \) |
| 43 | \( 1 + (-0.676 + 1.52i)T + (-28.9 - 31.8i)T^{2} \) |
| 47 | \( 1 + (-1.05 + 2.87i)T + (-35.7 - 30.4i)T^{2} \) |
| 53 | \( 1 + (-6.87 - 2.78i)T + (38.0 + 36.8i)T^{2} \) |
| 59 | \( 1 + (-2.32 - 4.45i)T + (-33.7 + 48.3i)T^{2} \) |
| 61 | \( 1 + (6.39 - 5.44i)T + (9.73 - 60.2i)T^{2} \) |
| 67 | \( 1 + (1.71 - 4.66i)T + (-51.0 - 43.4i)T^{2} \) |
| 71 | \( 1 + (-0.809 - 5.00i)T + (-67.3 + 22.3i)T^{2} \) |
| 73 | \( 1 + (-8.45 - 13.9i)T + (-33.7 + 64.7i)T^{2} \) |
| 79 | \( 1 + (-0.481 - 0.529i)T + (-7.58 + 78.6i)T^{2} \) |
| 83 | \( 1 + (-2.02 + 8.88i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-0.342 + 10.6i)T + (-88.8 - 5.70i)T^{2} \) |
| 97 | \( 1 + (3.35 + 4.80i)T + (-33.5 + 91.0i)T^{2} \) |
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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
−11.87151523130749699038697236901, −10.98216264405659207499967031437, −10.11489064639826316458440933046, −8.919607772101589708647370985987, −8.453210618130742770406641817602, −7.04262115440058417756307760675, −6.09469482082494236542572486488, −5.65328539812188520506504431308, −1.82725357673576165911590519369, −0.73006402595193429844391398028,
1.90757575618628590863910621158, 3.55635110508069162038798843591, 6.05586324242357819824319640133, 6.55516811379251778811522973005, 7.982025400824696919687702729286, 9.325353050739661844511591683665, 9.936371344769342376492203252148, 10.69553992020443522911366543092, 11.17316901832508278236770964585, 12.06272821000927038082103277364
