| L(s) = 1 | − 0.805·2-s − 3-s − 1.35·4-s + 1.06·5-s + 0.805·6-s − 0.203·7-s + 2.69·8-s + 9-s − 0.857·10-s + 11-s + 1.35·12-s + 0.801·13-s + 0.163·14-s − 1.06·15-s + 0.530·16-s − 0.376·17-s − 0.805·18-s + 2.62·19-s − 1.43·20-s + 0.203·21-s − 0.805·22-s + 2.93·23-s − 2.69·24-s − 3.86·25-s − 0.645·26-s − 27-s + 0.274·28-s + ⋯ |
| L(s) = 1 | − 0.569·2-s − 0.577·3-s − 0.675·4-s + 0.476·5-s + 0.328·6-s − 0.0767·7-s + 0.954·8-s + 0.333·9-s − 0.271·10-s + 0.301·11-s + 0.390·12-s + 0.222·13-s + 0.0436·14-s − 0.274·15-s + 0.132·16-s − 0.0913·17-s − 0.189·18-s + 0.601·19-s − 0.321·20-s + 0.0443·21-s − 0.171·22-s + 0.611·23-s − 0.550·24-s − 0.773·25-s − 0.126·26-s − 0.192·27-s + 0.0518·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9308249572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9308249572\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 0.805T + 2T^{2} \) |
| 5 | \( 1 - 1.06T + 5T^{2} \) |
| 7 | \( 1 + 0.203T + 7T^{2} \) |
| 13 | \( 1 - 0.801T + 13T^{2} \) |
| 17 | \( 1 + 0.376T + 17T^{2} \) |
| 19 | \( 1 - 2.62T + 19T^{2} \) |
| 23 | \( 1 - 2.93T + 23T^{2} \) |
| 29 | \( 1 - 0.907T + 29T^{2} \) |
| 31 | \( 1 - 2.07T + 31T^{2} \) |
| 37 | \( 1 + 2.14T + 37T^{2} \) |
| 41 | \( 1 + 1.83T + 41T^{2} \) |
| 43 | \( 1 + 2.25T + 43T^{2} \) |
| 47 | \( 1 + 0.685T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 - 7.73T + 59T^{2} \) |
| 67 | \( 1 + 7.59T + 67T^{2} \) |
| 71 | \( 1 + 9.23T + 71T^{2} \) |
| 73 | \( 1 - 7.49T + 73T^{2} \) |
| 79 | \( 1 - 7.41T + 79T^{2} \) |
| 83 | \( 1 + 8.55T + 83T^{2} \) |
| 89 | \( 1 + 2.97T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−9.161912991976350409287868521102, −8.561102036740882203465350728026, −7.63523452156112157052735719590, −6.86237854495135507096876576509, −5.92129499078810634627853484259, −5.18615325440123139855216701568, −4.37532505942466464039811707767, −3.37459199974488123360365570141, −1.84601055658500459611285127592, −0.75181304271230845439401636400,
0.75181304271230845439401636400, 1.84601055658500459611285127592, 3.37459199974488123360365570141, 4.37532505942466464039811707767, 5.18615325440123139855216701568, 5.92129499078810634627853484259, 6.86237854495135507096876576509, 7.63523452156112157052735719590, 8.561102036740882203465350728026, 9.161912991976350409287868521102
