| L(s) = 1 | − 4.89·2-s + 1.98·3-s + 15.9·4-s + 3.25·5-s − 9.70·6-s − 2.13·7-s − 38.7·8-s − 23.0·9-s − 15.9·10-s + 26.3·11-s + 31.6·12-s − 70.2·13-s + 10.4·14-s + 6.45·15-s + 62.2·16-s − 23.7·17-s + 112.·18-s + 158.·19-s + 51.8·20-s − 4.23·21-s − 128.·22-s + 62.3·23-s − 76.9·24-s − 114.·25-s + 343.·26-s − 99.3·27-s − 34.0·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s + 0.381·3-s + 1.99·4-s + 0.290·5-s − 0.660·6-s − 0.115·7-s − 1.71·8-s − 0.854·9-s − 0.503·10-s + 0.722·11-s + 0.760·12-s − 1.49·13-s + 0.199·14-s + 0.111·15-s + 0.973·16-s − 0.338·17-s + 1.47·18-s + 1.91·19-s + 0.579·20-s − 0.0440·21-s − 1.24·22-s + 0.565·23-s − 0.654·24-s − 0.915·25-s + 2.59·26-s − 0.708·27-s − 0.229·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 197 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 197 | \( 1 - 197T \) |
| good | 2 | \( 1 + 4.89T + 8T^{2} \) |
| 3 | \( 1 - 1.98T + 27T^{2} \) |
| 5 | \( 1 - 3.25T + 125T^{2} \) |
| 7 | \( 1 + 2.13T + 343T^{2} \) |
| 11 | \( 1 - 26.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 70.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 23.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 158.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 62.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 110.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 56.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 309.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 276.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 495.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 476.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 295.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 312.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 255.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 506.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 436.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 98.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.29e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 375.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 207.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 620.T + 9.12e5T^{2} \) |
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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
−11.44404461715669488493078777560, −10.09732965202947921549617072595, −9.476499311028062406321314335426, −8.755396691541534178916390137699, −7.64397803957944869428042817796, −6.87727336367694148282085811071, −5.38747698774695401012059218893, −3.10490795438607312113733698086, −1.77105108203841484372099299426, 0,
1.77105108203841484372099299426, 3.10490795438607312113733698086, 5.38747698774695401012059218893, 6.87727336367694148282085811071, 7.64397803957944869428042817796, 8.755396691541534178916390137699, 9.476499311028062406321314335426, 10.09732965202947921549617072595, 11.44404461715669488493078777560
