| L(s) = 1 | − 3-s + 5-s − 3·7-s − 2·9-s − 2·11-s + 13-s − 15-s − 3·17-s − 2·19-s + 3·21-s − 4·23-s − 4·25-s + 5·27-s + 2·29-s − 4·31-s + 2·33-s − 3·35-s + 5·37-s − 39-s − 12·41-s − 7·43-s − 2·45-s + 9·47-s + 2·49-s + 3·51-s + 4·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.13·7-s − 2/3·9-s − 0.603·11-s + 0.277·13-s − 0.258·15-s − 0.727·17-s − 0.458·19-s + 0.654·21-s − 0.834·23-s − 4/5·25-s + 0.962·27-s + 0.371·29-s − 0.718·31-s + 0.348·33-s − 0.507·35-s + 0.821·37-s − 0.160·39-s − 1.87·41-s − 1.06·43-s − 0.298·45-s + 1.31·47-s + 2/7·49-s + 0.420·51-s + 0.549·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−10.69730552679748697536416998477, −9.976912614651428361493366785064, −9.039528016718973273192662402272, −8.052297265441174562544922743090, −6.65408062467448396251062225874, −6.08119338191936199705626075673, −5.12564549383277447468893363985, −3.64849442843249490400769039530, −2.34351544305748892558428024728, 0,
2.34351544305748892558428024728, 3.64849442843249490400769039530, 5.12564549383277447468893363985, 6.08119338191936199705626075673, 6.65408062467448396251062225874, 8.052297265441174562544922743090, 9.039528016718973273192662402272, 9.976912614651428361493366785064, 10.69730552679748697536416998477
