| L(s) = 1 | − 2-s − 2.98·3-s + 4-s + 2.19·5-s + 2.98·6-s − 1.49·7-s − 8-s + 5.91·9-s − 2.19·10-s − 5.19·11-s − 2.98·12-s − 3.96·13-s + 1.49·14-s − 6.55·15-s + 16-s − 7.73·17-s − 5.91·18-s − 6.65·19-s + 2.19·20-s + 4.45·21-s + 5.19·22-s − 0.635·23-s + 2.98·24-s − 0.171·25-s + 3.96·26-s − 8.68·27-s − 1.49·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.72·3-s + 0.5·4-s + 0.982·5-s + 1.21·6-s − 0.563·7-s − 0.353·8-s + 1.97·9-s − 0.694·10-s − 1.56·11-s − 0.861·12-s − 1.09·13-s + 0.398·14-s − 1.69·15-s + 0.250·16-s − 1.87·17-s − 1.39·18-s − 1.52·19-s + 0.491·20-s + 0.971·21-s + 1.10·22-s − 0.132·23-s + 0.609·24-s − 0.0343·25-s + 0.777·26-s − 1.67·27-s − 0.281·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.008320651778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.008320651778\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
| good | 3 | \( 1 + 2.98T + 3T^{2} \) |
| 5 | \( 1 - 2.19T + 5T^{2} \) |
| 7 | \( 1 + 1.49T + 7T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 + 3.96T + 13T^{2} \) |
| 17 | \( 1 + 7.73T + 17T^{2} \) |
| 19 | \( 1 + 6.65T + 19T^{2} \) |
| 23 | \( 1 + 0.635T + 23T^{2} \) |
| 29 | \( 1 + 9.19T + 29T^{2} \) |
| 31 | \( 1 + 8.69T + 31T^{2} \) |
| 37 | \( 1 - 5.57T + 37T^{2} \) |
| 41 | \( 1 + 4.25T + 41T^{2} \) |
| 43 | \( 1 + 2.99T + 43T^{2} \) |
| 47 | \( 1 - 5.57T + 47T^{2} \) |
| 53 | \( 1 - 5.62T + 53T^{2} \) |
| 59 | \( 1 + 4.65T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 4.18T + 67T^{2} \) |
| 71 | \( 1 + 1.00T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 3.22T + 89T^{2} \) |
| 97 | \( 1 + 4.42T + 97T^{2} \) |
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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
−8.541564395559919055006061036603, −7.43542411872181221749026737584, −6.89893666892592264747437200609, −6.22059207737584068910364128739, −5.59509448580227067598997817619, −5.04850627365229755155741654048, −4.09198327316764204632079964895, −2.39353710383987047919300965353, −1.96307230561064088096761482052, −0.05825220483569670217681714652,
0.05825220483569670217681714652, 1.96307230561064088096761482052, 2.39353710383987047919300965353, 4.09198327316764204632079964895, 5.04850627365229755155741654048, 5.59509448580227067598997817619, 6.22059207737584068910364128739, 6.89893666892592264747437200609, 7.43542411872181221749026737584, 8.541564395559919055006061036603
