| L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s + 4·11-s + 2·13-s − 2·15-s − 6·17-s − 19-s + 21-s − 25-s + 27-s − 10·29-s − 8·31-s + 4·33-s − 2·35-s − 2·37-s + 2·39-s + 6·41-s − 4·43-s − 2·45-s + 8·47-s + 49-s − 6·51-s + 6·53-s − 8·55-s − 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.516·15-s − 1.45·17-s − 0.229·19-s + 0.218·21-s − 1/5·25-s + 0.192·27-s − 1.85·29-s − 1.43·31-s + 0.696·33-s − 0.338·35-s − 0.328·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s − 1.07·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.52575371914967734899270211735, −7.26283826961849506852997865294, −6.34405960679636745174264371605, −5.59578308199104929023134180855, −4.41838575600479152514036959003, −4.01788761669435895907720054899, −3.43571545368810704495867861827, −2.21597410635792504344440321860, −1.45880433129554543413888149463, 0,
1.45880433129554543413888149463, 2.21597410635792504344440321860, 3.43571545368810704495867861827, 4.01788761669435895907720054899, 4.41838575600479152514036959003, 5.59578308199104929023134180855, 6.34405960679636745174264371605, 7.26283826961849506852997865294, 7.52575371914967734899270211735
