| L(s) = 1 | − 2.46·2-s + 1.61·3-s + 4.08·4-s + 3.46·5-s − 3.99·6-s − 5.14·8-s − 0.381·9-s − 8.55·10-s + 6.60·12-s − 0.653·13-s + 5.60·15-s + 4.51·16-s − 1.13·17-s + 0.942·18-s − 6.07·19-s + 14.1·20-s − 6.66·23-s − 8.32·24-s + 7.01·25-s + 1.61·26-s − 5.47·27-s + 4.57·29-s − 13.8·30-s − 2.79·31-s − 0.854·32-s + 2.80·34-s − 1.56·36-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 0.934·3-s + 2.04·4-s + 1.55·5-s − 1.62·6-s − 1.81·8-s − 0.127·9-s − 2.70·10-s + 1.90·12-s − 0.181·13-s + 1.44·15-s + 1.12·16-s − 0.275·17-s + 0.222·18-s − 1.39·19-s + 3.16·20-s − 1.39·23-s − 1.69·24-s + 1.40·25-s + 0.316·26-s − 1.05·27-s + 0.849·29-s − 2.52·30-s − 0.502·31-s − 0.150·32-s + 0.481·34-s − 0.260·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 13 | \( 1 + 0.653T + 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 19 | \( 1 + 6.07T + 19T^{2} \) |
| 23 | \( 1 + 6.66T + 23T^{2} \) |
| 29 | \( 1 - 4.57T + 29T^{2} \) |
| 31 | \( 1 + 2.79T + 31T^{2} \) |
| 37 | \( 1 + 0.439T + 37T^{2} \) |
| 41 | \( 1 + 5.90T + 41T^{2} \) |
| 43 | \( 1 + 8.70T + 43T^{2} \) |
| 47 | \( 1 - 0.604T + 47T^{2} \) |
| 53 | \( 1 - 9.82T + 53T^{2} \) |
| 59 | \( 1 + 1.69T + 59T^{2} \) |
| 61 | \( 1 - 6.85T + 61T^{2} \) |
| 67 | \( 1 + 6.17T + 67T^{2} \) |
| 71 | \( 1 + 5.41T + 71T^{2} \) |
| 73 | \( 1 - 6.70T + 73T^{2} \) |
| 79 | \( 1 + 2.65T + 79T^{2} \) |
| 83 | \( 1 - 6.69T + 83T^{2} \) |
| 89 | \( 1 - 0.698T + 89T^{2} \) |
| 97 | \( 1 - 14.8T + 97T^{2} \) |
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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
−8.105898627843840772333543992897, −7.25608383604520273645426556925, −6.43230221883144536239673545350, −6.05798499805972919938161130191, −5.00504155549582620888964497391, −3.72457194786103346288113937539, −2.53228367236093518788572954993, −2.20063523926006758866616544433, −1.50192962034878921344735305803, 0,
1.50192962034878921344735305803, 2.20063523926006758866616544433, 2.53228367236093518788572954993, 3.72457194786103346288113937539, 5.00504155549582620888964497391, 6.05798499805972919938161130191, 6.43230221883144536239673545350, 7.25608383604520273645426556925, 8.105898627843840772333543992897
