| L(s) = 1 | + (−0.358 − 2.78i)2-s + (−1.19 − 0.441i)3-s + (−5.66 + 1.48i)4-s + (0.0438 − 0.0193i)5-s + (−0.796 + 3.49i)6-s + (−1.37 − 0.177i)7-s + (4.05 + 10.0i)8-s + (−1.04 − 0.887i)9-s + (−0.0696 − 0.114i)10-s + (1.38 + 2.12i)11-s + (7.44 + 0.718i)12-s + (−0.159 − 2.48i)13-s + 3.88i·14-s + (−0.0610 + 0.00392i)15-s + (16.2 − 9.12i)16-s + (−7.15 − 0.229i)17-s + ⋯ |
| L(s) = 1 | + (−0.253 − 1.96i)2-s + (−0.692 − 0.254i)3-s + (−2.83 + 0.742i)4-s + (0.0195 − 0.00867i)5-s + (−0.325 + 1.42i)6-s + (−0.520 − 0.0670i)7-s + (1.43 + 3.54i)8-s + (−0.347 − 0.295i)9-s + (−0.0220 − 0.0363i)10-s + (0.417 + 0.641i)11-s + (2.14 + 0.207i)12-s + (−0.0441 − 0.687i)13-s + 1.03i·14-s + (−0.0157 + 0.00101i)15-s + (4.05 − 2.28i)16-s + (−1.73 − 0.0556i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.359 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.134404 + 0.0922043i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.134404 + 0.0922043i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 + (12.9 - 5.32i)T \) |
| good | 2 | \( 1 + (0.358 + 2.78i)T + (-1.93 + 0.507i)T^{2} \) |
| 3 | \( 1 + (1.19 + 0.441i)T + (2.28 + 1.94i)T^{2} \) |
| 5 | \( 1 + (-0.0438 + 0.0193i)T + (3.36 - 3.70i)T^{2} \) |
| 7 | \( 1 + (1.37 + 0.177i)T + (6.77 + 1.77i)T^{2} \) |
| 11 | \( 1 + (-1.38 - 2.12i)T + (-4.45 + 10.0i)T^{2} \) |
| 13 | \( 1 + (0.159 + 2.48i)T + (-12.8 + 1.66i)T^{2} \) |
| 17 | \( 1 + (7.15 + 0.229i)T + (16.9 + 1.08i)T^{2} \) |
| 19 | \( 1 + (1.48 - 1.86i)T + (-4.22 - 18.5i)T^{2} \) |
| 23 | \( 1 + (3.60 + 1.19i)T + (18.4 + 13.7i)T^{2} \) |
| 29 | \( 1 + (-0.442 + 4.58i)T + (-28.4 - 5.54i)T^{2} \) |
| 31 | \( 1 + (-4.21 + 4.35i)T + (-0.993 - 30.9i)T^{2} \) |
| 37 | \( 1 + (5.37 + 3.02i)T + (19.1 + 31.6i)T^{2} \) |
| 41 | \( 1 + (0.165 - 5.14i)T + (-40.9 - 2.62i)T^{2} \) |
| 43 | \( 1 + (-1.07 + 0.697i)T + (17.4 - 39.3i)T^{2} \) |
| 47 | \( 1 + (1.07 + 1.53i)T + (-16.2 + 44.1i)T^{2} \) |
| 53 | \( 1 + (9.14 + 1.78i)T + (49.1 + 19.8i)T^{2} \) |
| 59 | \( 1 + (1.15 - 1.90i)T + (-27.2 - 52.3i)T^{2} \) |
| 61 | \( 1 + (0.791 + 2.15i)T + (-46.4 + 39.5i)T^{2} \) |
| 67 | \( 1 + (-10.3 + 7.25i)T + (23.1 - 62.8i)T^{2} \) |
| 71 | \( 1 + (-3.07 + 3.61i)T + (-11.3 - 70.0i)T^{2} \) |
| 73 | \( 1 + (-7.42 + 13.1i)T + (-37.8 - 62.4i)T^{2} \) |
| 79 | \( 1 + (-13.4 - 5.96i)T + (53.1 + 58.4i)T^{2} \) |
| 83 | \( 1 + (10.3 + 13.0i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (3.66 + 3.77i)T + (-2.85 + 88.9i)T^{2} \) |
| 97 | \( 1 + (0.971 - 1.86i)T + (-55.4 - 79.5i)T^{2} \) |
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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
−11.63197630377756812412650363282, −10.94835856087882253106003686433, −9.935248242512608958444985619263, −9.195212188349906765809658151826, −8.047776058189067488954312551295, −6.24050978181463555962373972777, −4.76007411059383454465213562678, −3.58225130195631563945093215326, −2.09393820485516562332082652871, −0.16599833274372428335910684646,
4.16315713237469392724847254101, 5.17169034581936715009066666851, 6.34074321263398933467531480803, 6.71768822049406806883177716351, 8.273550813166706405378970464002, 8.935775787543297524645305309580, 9.990463907999834385115959546961, 11.14605868229078626495548021536, 12.61353212316450594530345094716, 13.85381787052922845275614156767
