| L(s) = 1 | − 0.690·2-s + 1.58·3-s − 1.52·4-s + 3.69·5-s − 1.09·6-s + 2.15·7-s + 2.43·8-s − 0.492·9-s − 2.55·10-s − 2.39·11-s − 2.41·12-s + 2.73·13-s − 1.48·14-s + 5.85·15-s + 1.36·16-s + 17-s + 0.339·18-s − 1.26·19-s − 5.63·20-s + 3.41·21-s + 1.65·22-s + 9.19·23-s + 3.85·24-s + 8.68·25-s − 1.88·26-s − 5.53·27-s − 3.28·28-s + ⋯ |
| L(s) = 1 | − 0.488·2-s + 0.914·3-s − 0.761·4-s + 1.65·5-s − 0.446·6-s + 0.814·7-s + 0.860·8-s − 0.164·9-s − 0.807·10-s − 0.720·11-s − 0.696·12-s + 0.757·13-s − 0.397·14-s + 1.51·15-s + 0.341·16-s + 0.242·17-s + 0.0801·18-s − 0.289·19-s − 1.25·20-s + 0.744·21-s + 0.351·22-s + 1.91·23-s + 0.786·24-s + 1.73·25-s − 0.369·26-s − 1.06·27-s − 0.620·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.029209211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.029209211\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 + 0.690T + 2T^{2} \) |
| 3 | \( 1 - 1.58T + 3T^{2} \) |
| 5 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 - 2.15T + 7T^{2} \) |
| 11 | \( 1 + 2.39T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 19 | \( 1 + 1.26T + 19T^{2} \) |
| 23 | \( 1 - 9.19T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 + 6.63T + 31T^{2} \) |
| 37 | \( 1 - 3.23T + 37T^{2} \) |
| 41 | \( 1 - 0.0108T + 41T^{2} \) |
| 43 | \( 1 + 0.666T + 43T^{2} \) |
| 47 | \( 1 + 8.17T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 6.43T + 67T^{2} \) |
| 71 | \( 1 - 4.64T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 7.75T + 79T^{2} \) |
| 83 | \( 1 - 4.55T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 5.58T + 97T^{2} \) |
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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
−9.505341544785123623336676621226, −9.319201641951232096388689809281, −8.386288521653323419562562009938, −7.938140220608146462418913139276, −6.64319347887618625911778842731, −5.42031761764645203814792844552, −4.98310717007625358799622693803, −3.47761171339462976991989605317, −2.32573706352645706917236043963, −1.32551172003723575470570603034,
1.32551172003723575470570603034, 2.32573706352645706917236043963, 3.47761171339462976991989605317, 4.98310717007625358799622693803, 5.42031761764645203814792844552, 6.64319347887618625911778842731, 7.938140220608146462418913139276, 8.386288521653323419562562009938, 9.319201641951232096388689809281, 9.505341544785123623336676621226
