| L(s) = 1 | − 2-s + 4-s + 2.46·7-s − 8-s − 0.728·11-s − 6.23·13-s − 2.46·14-s + 16-s − 0.563·17-s − 19-s + 0.728·22-s − 4.63·23-s + 6.23·26-s + 2.46·28-s − 10.2·29-s + 6.06·31-s − 32-s + 0.563·34-s − 5.72·37-s + 38-s − 4.79·41-s + 8.06·43-s − 0.728·44-s + 4.63·46-s − 8.12·47-s − 0.900·49-s − 6.23·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.933·7-s − 0.353·8-s − 0.219·11-s − 1.72·13-s − 0.660·14-s + 0.250·16-s − 0.136·17-s − 0.229·19-s + 0.155·22-s − 0.966·23-s + 1.22·26-s + 0.466·28-s − 1.89·29-s + 1.08·31-s − 0.176·32-s + 0.0965·34-s − 0.941·37-s + 0.162·38-s − 0.748·41-s + 1.23·43-s − 0.109·44-s + 0.683·46-s − 1.18·47-s − 0.128·49-s − 0.864·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.039860584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.039860584\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 2.46T + 7T^{2} \) |
| 11 | \( 1 + 0.728T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 + 0.563T + 17T^{2} \) |
| 23 | \( 1 + 4.63T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 6.06T + 31T^{2} \) |
| 37 | \( 1 + 5.72T + 37T^{2} \) |
| 41 | \( 1 + 4.79T + 41T^{2} \) |
| 43 | \( 1 - 8.06T + 43T^{2} \) |
| 47 | \( 1 + 8.12T + 47T^{2} \) |
| 53 | \( 1 - 1.53T + 53T^{2} \) |
| 59 | \( 1 - 5.76T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 + 4.09T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 7.85T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−7.975239392201194734833546528512, −7.20711779628335621987261197324, −6.67131918531023650799606680141, −5.62402299051102079954756523583, −5.09002319492617793987502708246, −4.33982975164361804492220988575, −3.38425539607140498469565534189, −2.21501035721447891472946580252, −1.94696767510668984163891284021, −0.53093777693241158996295069653,
0.53093777693241158996295069653, 1.94696767510668984163891284021, 2.21501035721447891472946580252, 3.38425539607140498469565534189, 4.33982975164361804492220988575, 5.09002319492617793987502708246, 5.62402299051102079954756523583, 6.67131918531023650799606680141, 7.20711779628335621987261197324, 7.975239392201194734833546528512
