| L(s) = 1 | + 2.66·2-s + 3-s + 5.10·4-s − 5-s + 2.66·6-s + 7-s + 8.27·8-s + 9-s − 2.66·10-s + 11-s + 5.10·12-s − 3.60·13-s + 2.66·14-s − 15-s + 11.8·16-s − 2.62·17-s + 2.66·18-s − 2.10·19-s − 5.10·20-s + 21-s + 2.66·22-s + 1.22·23-s + 8.27·24-s + 25-s − 9.60·26-s + 27-s + 5.10·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 0.577·3-s + 2.55·4-s − 0.447·5-s + 1.08·6-s + 0.377·7-s + 2.92·8-s + 0.333·9-s − 0.842·10-s + 0.301·11-s + 1.47·12-s − 0.999·13-s + 0.712·14-s − 0.258·15-s + 2.95·16-s − 0.637·17-s + 0.628·18-s − 0.482·19-s − 1.14·20-s + 0.218·21-s + 0.568·22-s + 0.255·23-s + 1.68·24-s + 0.200·25-s − 1.88·26-s + 0.192·27-s + 0.964·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.903242230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.903242230\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 13 | \( 1 + 3.60T + 13T^{2} \) |
| 17 | \( 1 + 2.62T + 17T^{2} \) |
| 19 | \( 1 + 2.10T + 19T^{2} \) |
| 23 | \( 1 - 1.22T + 23T^{2} \) |
| 29 | \( 1 - 5.23T + 29T^{2} \) |
| 31 | \( 1 - 2.27T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 4.20T + 41T^{2} \) |
| 43 | \( 1 + 4.70T + 43T^{2} \) |
| 47 | \( 1 - 4.13T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 8.30T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 9.31T + 83T^{2} \) |
| 89 | \( 1 + 7.25T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−10.09664382225540915971933754582, −8.790269676981491246001084508526, −7.87761842901164261206654683938, −7.00168520555940840472458555126, −6.44768770791701489749478225696, −5.12701870544903204030314817238, −4.61837879106746725480508806681, −3.72535845293637296989215737776, −2.82136079909335777599720043105, −1.85852278820446617148957629358,
1.85852278820446617148957629358, 2.82136079909335777599720043105, 3.72535845293637296989215737776, 4.61837879106746725480508806681, 5.12701870544903204030314817238, 6.44768770791701489749478225696, 7.00168520555940840472458555126, 7.87761842901164261206654683938, 8.790269676981491246001084508526, 10.09664382225540915971933754582
