| L(s) = 1 | + 1.54·2-s + 2.81·3-s + 0.398·4-s + 0.334·5-s + 4.35·6-s − 4.24·7-s − 2.47·8-s + 4.90·9-s + 0.518·10-s + 0.915·11-s + 1.12·12-s + 4.81·13-s − 6.57·14-s + 0.941·15-s − 4.63·16-s − 5.38·17-s + 7.59·18-s − 4.34·19-s + 0.133·20-s − 11.9·21-s + 1.41·22-s + 8.10·23-s − 6.97·24-s − 4.88·25-s + 7.45·26-s + 5.34·27-s − 1.69·28-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 1.62·3-s + 0.199·4-s + 0.149·5-s + 1.77·6-s − 1.60·7-s − 0.876·8-s + 1.63·9-s + 0.164·10-s + 0.276·11-s + 0.323·12-s + 1.33·13-s − 1.75·14-s + 0.243·15-s − 1.15·16-s − 1.30·17-s + 1.78·18-s − 0.997·19-s + 0.0298·20-s − 2.60·21-s + 0.302·22-s + 1.69·23-s − 1.42·24-s − 0.977·25-s + 1.46·26-s + 1.02·27-s − 0.319·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.702826397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.702826397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 241 | \( 1 - T \) |
| good | 2 | \( 1 - 1.54T + 2T^{2} \) |
| 3 | \( 1 - 2.81T + 3T^{2} \) |
| 5 | \( 1 - 0.334T + 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 - 0.915T + 11T^{2} \) |
| 13 | \( 1 - 4.81T + 13T^{2} \) |
| 17 | \( 1 + 5.38T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 23 | \( 1 - 8.10T + 23T^{2} \) |
| 29 | \( 1 + 6.45T + 29T^{2} \) |
| 31 | \( 1 - 10.7T + 31T^{2} \) |
| 37 | \( 1 - 5.16T + 37T^{2} \) |
| 41 | \( 1 + 0.612T + 41T^{2} \) |
| 43 | \( 1 + 1.85T + 43T^{2} \) |
| 47 | \( 1 + 2.21T + 47T^{2} \) |
| 53 | \( 1 - 0.00846T + 53T^{2} \) |
| 59 | \( 1 - 8.85T + 59T^{2} \) |
| 61 | \( 1 - 3.78T + 61T^{2} \) |
| 67 | \( 1 - 4.67T + 67T^{2} \) |
| 71 | \( 1 - 1.48T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 17.6T + 79T^{2} \) |
| 83 | \( 1 - 7.45T + 83T^{2} \) |
| 89 | \( 1 + 0.520T + 89T^{2} \) |
| 97 | \( 1 + 6.33T + 97T^{2} \) |
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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
−12.90389229477392581509870676619, −11.36416778378770610673450445351, −9.896701957944242820230079154929, −9.061702660070623275858419629653, −8.532666071906962867081280326261, −6.83414834217148042912106686480, −6.10425309379006414572978486043, −4.26327692610701359505282660648, −3.50045960627864030408212168559, −2.57307631818207375293336840940,
2.57307631818207375293336840940, 3.50045960627864030408212168559, 4.26327692610701359505282660648, 6.10425309379006414572978486043, 6.83414834217148042912106686480, 8.532666071906962867081280326261, 9.061702660070623275858419629653, 9.896701957944242820230079154929, 11.36416778378770610673450445351, 12.90389229477392581509870676619
