| L(s) = 1 | + 0.525·2-s − 1.36·3-s − 1.72·4-s − 0.718·6-s + 0.426·7-s − 1.95·8-s − 1.13·9-s − 1.05·11-s + 2.35·12-s − 2.19·13-s + 0.224·14-s + 2.42·16-s − 0.594·18-s + 2.52·19-s − 0.583·21-s − 0.551·22-s − 3.45·23-s + 2.67·24-s − 1.15·26-s + 5.64·27-s − 0.735·28-s − 6.88·29-s + 1.94·31-s + 5.18·32-s + 1.43·33-s + 1.95·36-s + 1.86·37-s + ⋯ |
| L(s) = 1 | + 0.371·2-s − 0.789·3-s − 0.862·4-s − 0.293·6-s + 0.161·7-s − 0.691·8-s − 0.377·9-s − 0.316·11-s + 0.680·12-s − 0.608·13-s + 0.0599·14-s + 0.605·16-s − 0.140·18-s + 0.578·19-s − 0.127·21-s − 0.117·22-s − 0.720·23-s + 0.545·24-s − 0.226·26-s + 1.08·27-s − 0.139·28-s − 1.27·29-s + 0.349·31-s + 0.916·32-s + 0.249·33-s + 0.325·36-s + 0.306·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5229724018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5229724018\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.525T + 2T^{2} \) |
| 3 | \( 1 + 1.36T + 3T^{2} \) |
| 7 | \( 1 - 0.426T + 7T^{2} \) |
| 11 | \( 1 + 1.05T + 11T^{2} \) |
| 13 | \( 1 + 2.19T + 13T^{2} \) |
| 19 | \( 1 - 2.52T + 19T^{2} \) |
| 23 | \( 1 + 3.45T + 23T^{2} \) |
| 29 | \( 1 + 6.88T + 29T^{2} \) |
| 31 | \( 1 - 1.94T + 31T^{2} \) |
| 37 | \( 1 - 1.86T + 37T^{2} \) |
| 41 | \( 1 + 9.98T + 41T^{2} \) |
| 43 | \( 1 + 8.66T + 43T^{2} \) |
| 47 | \( 1 + 2.79T + 47T^{2} \) |
| 53 | \( 1 - 7.96T + 53T^{2} \) |
| 59 | \( 1 + 8.61T + 59T^{2} \) |
| 61 | \( 1 + 0.659T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 3.86T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 + 2.51T + 83T^{2} \) |
| 89 | \( 1 + 5.82T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.948455050531724981687006900571, −7.19572235437899058989042967536, −6.22980555701311295959706767314, −5.73758582836153596566124392608, −5.00035855675696260728102676365, −4.67675722562914335209785396173, −3.60303998351056108969884520348, −2.94757081935760284001501633627, −1.70597259851932831711558368028, −0.35629072027363159188207827227,
0.35629072027363159188207827227, 1.70597259851932831711558368028, 2.94757081935760284001501633627, 3.60303998351056108969884520348, 4.67675722562914335209785396173, 5.00035855675696260728102676365, 5.73758582836153596566124392608, 6.22980555701311295959706767314, 7.19572235437899058989042967536, 7.948455050531724981687006900571
