L(s) = 1 | − 2.28·2-s + 3-s + 3.20·4-s + 2.67·5-s − 2.28·6-s − 3.60·7-s − 2.73·8-s + 9-s − 6.10·10-s − 4.75·11-s + 3.20·12-s − 2.82·13-s + 8.22·14-s + 2.67·15-s − 0.154·16-s + 17-s − 2.28·18-s + 7.19·19-s + 8.56·20-s − 3.60·21-s + 10.8·22-s − 7.90·23-s − 2.73·24-s + 2.15·25-s + 6.44·26-s + 27-s − 11.5·28-s + ⋯ |
L(s) = 1 | − 1.61·2-s + 0.577·3-s + 1.60·4-s + 1.19·5-s − 0.931·6-s − 1.36·7-s − 0.968·8-s + 0.333·9-s − 1.92·10-s − 1.43·11-s + 0.924·12-s − 0.784·13-s + 2.19·14-s + 0.690·15-s − 0.0385·16-s + 0.242·17-s − 0.537·18-s + 1.65·19-s + 1.91·20-s − 0.787·21-s + 2.31·22-s − 1.64·23-s − 0.559·24-s + 0.430·25-s + 1.26·26-s + 0.192·27-s − 2.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8499871111\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8499871111\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 5 | \( 1 - 2.67T + 5T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 + 4.75T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 19 | \( 1 - 7.19T + 19T^{2} \) |
| 23 | \( 1 + 7.90T + 23T^{2} \) |
| 29 | \( 1 - 3.76T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 3.99T + 41T^{2} \) |
| 43 | \( 1 + 2.71T + 43T^{2} \) |
| 47 | \( 1 - 7.28T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 - 1.09T + 59T^{2} \) |
| 61 | \( 1 - 8.22T + 61T^{2} \) |
| 67 | \( 1 + 1.58T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 5.52T + 89T^{2} \) |
| 97 | \( 1 - 9.40T + 97T^{2} \) |
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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
−8.487352058013038136003461876181, −7.87167451096625965414838179757, −7.24648119835898934206292185876, −6.48063071240688070324261822368, −5.76076398876110264026626495005, −4.85072564216263164320428064330, −3.28928490115503818198769788102, −2.60823023464598762792285493954, −1.95786115453075825286322617952, −0.62632184626964413810846970756,
0.62632184626964413810846970756, 1.95786115453075825286322617952, 2.60823023464598762792285493954, 3.28928490115503818198769788102, 4.85072564216263164320428064330, 5.76076398876110264026626495005, 6.48063071240688070324261822368, 7.24648119835898934206292185876, 7.87167451096625965414838179757, 8.487352058013038136003461876181