| L(s) = 1 | − 1.73·2-s + 1.01·4-s + 2.63·5-s − 2.01·7-s + 1.70·8-s − 4.57·10-s + 3.39·11-s + 5.63·13-s + 3.49·14-s − 4.99·16-s − 0.866·17-s + 2.46·19-s + 2.67·20-s − 5.89·22-s + 6.37·23-s + 1.92·25-s − 9.79·26-s − 2.04·28-s + 4.52·29-s − 3.51·31-s + 5.26·32-s + 1.50·34-s − 5.29·35-s − 5.19·37-s − 4.28·38-s + 4.49·40-s − 1.35·41-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.508·4-s + 1.17·5-s − 0.759·7-s + 0.603·8-s − 1.44·10-s + 1.02·11-s + 1.56·13-s + 0.933·14-s − 1.24·16-s − 0.210·17-s + 0.565·19-s + 0.598·20-s − 1.25·22-s + 1.33·23-s + 0.385·25-s − 1.91·26-s − 0.386·28-s + 0.839·29-s − 0.631·31-s + 0.931·32-s + 0.258·34-s − 0.894·35-s − 0.853·37-s − 0.694·38-s + 0.710·40-s − 0.211·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.240303174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.240303174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 5 | \( 1 - 2.63T + 5T^{2} \) |
| 7 | \( 1 + 2.01T + 7T^{2} \) |
| 11 | \( 1 - 3.39T + 11T^{2} \) |
| 13 | \( 1 - 5.63T + 13T^{2} \) |
| 17 | \( 1 + 0.866T + 17T^{2} \) |
| 19 | \( 1 - 2.46T + 19T^{2} \) |
| 23 | \( 1 - 6.37T + 23T^{2} \) |
| 29 | \( 1 - 4.52T + 29T^{2} \) |
| 31 | \( 1 + 3.51T + 31T^{2} \) |
| 37 | \( 1 + 5.19T + 37T^{2} \) |
| 41 | \( 1 + 1.35T + 41T^{2} \) |
| 43 | \( 1 - 8.49T + 43T^{2} \) |
| 47 | \( 1 - 9.44T + 47T^{2} \) |
| 53 | \( 1 + 9.71T + 53T^{2} \) |
| 59 | \( 1 + 6.03T + 59T^{2} \) |
| 61 | \( 1 - 4.45T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 - 3.01T + 71T^{2} \) |
| 73 | \( 1 + 0.255T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 16.8T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 0.273T + 97T^{2} \) |
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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
−9.215346848300112174187822251490, −8.700361676308640167163511130896, −7.63369407186685769921485397581, −6.67402879730091914292647849019, −6.26945555834336181035110482977, −5.28857168404428725481032051845, −4.08326628800448549368667722083, −3.04668305644489591501918654679, −1.71914014257974484895952677401, −0.966451702440892308446168591401,
0.966451702440892308446168591401, 1.71914014257974484895952677401, 3.04668305644489591501918654679, 4.08326628800448549368667722083, 5.28857168404428725481032051845, 6.26945555834336181035110482977, 6.67402879730091914292647849019, 7.63369407186685769921485397581, 8.700361676308640167163511130896, 9.215346848300112174187822251490
