L(s) = 1 | − 0.656·2-s − 1.65·3-s − 1.56·4-s + 1.08·6-s + 2.34·8-s − 0.255·9-s + 1.82·11-s + 2.59·12-s + 2.59·13-s + 1.59·16-s − 0.568·17-s + 0.167·18-s + 2.74·19-s − 1.19·22-s − 5.48·23-s − 3.88·24-s − 5·25-s − 1.70·26-s + 5.39·27-s − 1.31·29-s − 9.64·31-s − 5.73·32-s − 3.02·33-s + 0.373·34-s + 0.401·36-s − 3.40·37-s − 1.80·38-s + ⋯ |
L(s) = 1 | − 0.464·2-s − 0.956·3-s − 0.784·4-s + 0.444·6-s + 0.828·8-s − 0.0852·9-s + 0.550·11-s + 0.750·12-s + 0.720·13-s + 0.399·16-s − 0.137·17-s + 0.0395·18-s + 0.629·19-s − 0.255·22-s − 1.14·23-s − 0.792·24-s − 25-s − 0.334·26-s + 1.03·27-s − 0.243·29-s − 1.73·31-s − 1.01·32-s − 0.526·33-s + 0.0640·34-s + 0.0668·36-s − 0.559·37-s − 0.292·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5946172332\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5946172332\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 + 0.656T + 2T^{2} \) |
| 3 | \( 1 + 1.65T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 1.82T + 11T^{2} \) |
| 13 | \( 1 - 2.59T + 13T^{2} \) |
| 17 | \( 1 + 0.568T + 17T^{2} \) |
| 19 | \( 1 - 2.74T + 19T^{2} \) |
| 23 | \( 1 + 5.48T + 23T^{2} \) |
| 29 | \( 1 + 1.31T + 29T^{2} \) |
| 31 | \( 1 + 9.64T + 31T^{2} \) |
| 37 | \( 1 + 3.40T + 37T^{2} \) |
| 43 | \( 1 - 2.56T + 43T^{2} \) |
| 47 | \( 1 - 3.73T + 47T^{2} \) |
| 53 | \( 1 - 1.82T + 53T^{2} \) |
| 59 | \( 1 - 6.33T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 6.51T + 71T^{2} \) |
| 73 | \( 1 + 5.19T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 + 1.13T + 83T^{2} \) |
| 89 | \( 1 - 2.59T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−9.128093667079158757980263425504, −8.514597161438106273538148084324, −7.67536256495095490403583237224, −6.78902796673198617108705211862, −5.74101432670967565354028464396, −5.41582365237570744644510117108, −4.22248495341417712037576700795, −3.58195727433523434818337767375, −1.82744876008328932777988601048, −0.58950026196098288091025239744,
0.58950026196098288091025239744, 1.82744876008328932777988601048, 3.58195727433523434818337767375, 4.22248495341417712037576700795, 5.41582365237570744644510117108, 5.74101432670967565354028464396, 6.78902796673198617108705211862, 7.67536256495095490403583237224, 8.514597161438106273538148084324, 9.128093667079158757980263425504