| L(s) = 1 | − 2-s + 3.36·3-s + 4-s − 3.01·5-s − 3.36·6-s + 1.60·7-s − 8-s + 8.31·9-s + 3.01·10-s − 4.74·11-s + 3.36·12-s − 1.60·14-s − 10.1·15-s + 16-s − 3.15·17-s − 8.31·18-s − 19-s − 3.01·20-s + 5.40·21-s + 4.74·22-s − 4.43·23-s − 3.36·24-s + 4.10·25-s + 17.8·27-s + 1.60·28-s + 1.98·29-s + 10.1·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.94·3-s + 0.5·4-s − 1.34·5-s − 1.37·6-s + 0.607·7-s − 0.353·8-s + 2.77·9-s + 0.954·10-s − 1.43·11-s + 0.971·12-s − 0.429·14-s − 2.62·15-s + 0.250·16-s − 0.764·17-s − 1.96·18-s − 0.229·19-s − 0.674·20-s + 1.17·21-s + 1.01·22-s − 0.925·23-s − 0.686·24-s + 0.821·25-s + 3.44·27-s + 0.303·28-s + 0.368·29-s + 1.85·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.262524355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.262524355\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 3.36T + 3T^{2} \) |
| 5 | \( 1 + 3.01T + 5T^{2} \) |
| 7 | \( 1 - 1.60T + 7T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 17 | \( 1 + 3.15T + 17T^{2} \) |
| 23 | \( 1 + 4.43T + 23T^{2} \) |
| 29 | \( 1 - 1.98T + 29T^{2} \) |
| 31 | \( 1 - 3.75T + 31T^{2} \) |
| 37 | \( 1 - 8.45T + 37T^{2} \) |
| 41 | \( 1 - 6.86T + 41T^{2} \) |
| 43 | \( 1 - 4.39T + 43T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + 9.81T + 53T^{2} \) |
| 59 | \( 1 - 5.56T + 59T^{2} \) |
| 61 | \( 1 - 7.01T + 61T^{2} \) |
| 67 | \( 1 + 4.31T + 67T^{2} \) |
| 71 | \( 1 + 0.512T + 71T^{2} \) |
| 73 | \( 1 - 0.764T + 73T^{2} \) |
| 79 | \( 1 + 1.84T + 79T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 6.85T + 89T^{2} \) |
| 97 | \( 1 - 0.986T + 97T^{2} \) |
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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
−8.005915419548859065343277915048, −7.70296290098487958070668185503, −7.24635390413688682139376652207, −6.17156300823150462569261445782, −4.72198332842682865826386125520, −4.26037333789258908618299334015, −3.44783845868400450874110614910, −2.56610656498036403911784423343, −2.15192351321032080429954334750, −0.77245776205247139677564933178,
0.77245776205247139677564933178, 2.15192351321032080429954334750, 2.56610656498036403911784423343, 3.44783845868400450874110614910, 4.26037333789258908618299334015, 4.72198332842682865826386125520, 6.17156300823150462569261445782, 7.24635390413688682139376652207, 7.70296290098487958070668185503, 8.005915419548859065343277915048
