| L(s) = 1 | + 2-s + 1.91·3-s + 4-s − 5-s + 1.91·6-s + 7-s + 8-s + 0.683·9-s − 10-s + 1.91·12-s + 0.869·13-s + 14-s − 1.91·15-s + 16-s + 4.15·17-s + 0.683·18-s − 7.56·19-s − 20-s + 1.91·21-s − 2·23-s + 1.91·24-s + 25-s + 0.869·26-s − 4.44·27-s + 28-s + 1.19·29-s − 1.91·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.10·3-s + 0.5·4-s − 0.447·5-s + 0.783·6-s + 0.377·7-s + 0.353·8-s + 0.227·9-s − 0.316·10-s + 0.554·12-s + 0.241·13-s + 0.267·14-s − 0.495·15-s + 0.250·16-s + 1.00·17-s + 0.160·18-s − 1.73·19-s − 0.223·20-s + 0.418·21-s − 0.417·23-s + 0.391·24-s + 0.200·25-s + 0.170·26-s − 0.855·27-s + 0.188·28-s + 0.222·29-s − 0.350·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.918667909\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.918667909\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.91T + 3T^{2} \) |
| 13 | \( 1 - 0.869T + 13T^{2} \) |
| 17 | \( 1 - 4.15T + 17T^{2} \) |
| 19 | \( 1 + 7.56T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 1.19T + 29T^{2} \) |
| 31 | \( 1 - 7.70T + 31T^{2} \) |
| 37 | \( 1 - 6.81T + 37T^{2} \) |
| 41 | \( 1 - 9.86T + 41T^{2} \) |
| 43 | \( 1 - 6.62T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 + 1.57T + 53T^{2} \) |
| 59 | \( 1 + 7.43T + 59T^{2} \) |
| 61 | \( 1 - 7.73T + 61T^{2} \) |
| 67 | \( 1 - 3.56T + 67T^{2} \) |
| 71 | \( 1 + 2.23T + 71T^{2} \) |
| 73 | \( 1 + 17.0T + 73T^{2} \) |
| 79 | \( 1 - 3.13T + 79T^{2} \) |
| 83 | \( 1 - 8.22T + 83T^{2} \) |
| 89 | \( 1 + 9.89T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.68714679305717208231123400405, −7.39747818904429612463911215141, −6.13552395417871940419082579343, −5.93364656027160240385681956040, −4.66764473060443797495167573215, −4.21619281443108146878421706784, −3.53279989610734530522788604446, −2.67346039142872270468538284733, −2.18494701443797804958311475323, −0.933933928100366358608002462078,
0.933933928100366358608002462078, 2.18494701443797804958311475323, 2.67346039142872270468538284733, 3.53279989610734530522788604446, 4.21619281443108146878421706784, 4.66764473060443797495167573215, 5.93364656027160240385681956040, 6.13552395417871940419082579343, 7.39747818904429612463911215141, 7.68714679305717208231123400405
