| L(s) = 1 | − 2.38·2-s + 2.43·3-s + 3.66·4-s + 5-s − 5.80·6-s + 7-s − 3.97·8-s + 2.93·9-s − 2.38·10-s + 0.150·11-s + 8.93·12-s − 2.46·13-s − 2.38·14-s + 2.43·15-s + 2.12·16-s − 0.276·17-s − 6.98·18-s + 1.36·19-s + 3.66·20-s + 2.43·21-s − 0.359·22-s − 3.21·23-s − 9.68·24-s + 25-s + 5.86·26-s − 0.157·27-s + 3.66·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.40·3-s + 1.83·4-s + 0.447·5-s − 2.36·6-s + 0.377·7-s − 1.40·8-s + 0.978·9-s − 0.752·10-s + 0.0454·11-s + 2.58·12-s − 0.683·13-s − 0.636·14-s + 0.629·15-s + 0.531·16-s − 0.0670·17-s − 1.64·18-s + 0.313·19-s + 0.820·20-s + 0.531·21-s − 0.0765·22-s − 0.669·23-s − 1.97·24-s + 0.200·25-s + 1.15·26-s − 0.0304·27-s + 0.693·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 2.38T + 2T^{2} \) |
| 3 | \( 1 - 2.43T + 3T^{2} \) |
| 11 | \( 1 - 0.150T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 + 0.276T + 17T^{2} \) |
| 19 | \( 1 - 1.36T + 19T^{2} \) |
| 23 | \( 1 + 3.21T + 23T^{2} \) |
| 29 | \( 1 - 2.97T + 29T^{2} \) |
| 31 | \( 1 + 7.87T + 31T^{2} \) |
| 37 | \( 1 - 1.19T + 37T^{2} \) |
| 41 | \( 1 + 7.87T + 41T^{2} \) |
| 43 | \( 1 + 5.90T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 5.17T + 53T^{2} \) |
| 59 | \( 1 + 4.75T + 59T^{2} \) |
| 61 | \( 1 + 7.74T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 4.25T + 71T^{2} \) |
| 73 | \( 1 - 1.43T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 - 10.2T + 89T^{2} \) |
| 97 | \( 1 + 15.9T + 97T^{2} \) |
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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
−7.84349946776416060718315761366, −7.14832877291431148217312108481, −6.58517385446391290310084167662, −5.52368713192893648927995748569, −4.59574126676099131380767443004, −3.50100134087306887736655656780, −2.74254309608895901342434785340, −1.96800590629465876686635566042, −1.48876454978401115125965572724, 0,
1.48876454978401115125965572724, 1.96800590629465876686635566042, 2.74254309608895901342434785340, 3.50100134087306887736655656780, 4.59574126676099131380767443004, 5.52368713192893648927995748569, 6.58517385446391290310084167662, 7.14832877291431148217312108481, 7.84349946776416060718315761366
