| L(s) = 1 | − 2·3-s + 3·5-s − 7-s + 9-s − 13-s − 6·15-s − 2·17-s − 5·19-s + 2·21-s − 23-s + 4·25-s + 4·27-s − 5·29-s + 3·31-s − 3·35-s − 6·37-s + 2·39-s − 2·41-s − 43-s + 3·45-s − 3·47-s + 49-s + 4·51-s + 11·53-s + 10·57-s + 8·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.277·13-s − 1.54·15-s − 0.485·17-s − 1.14·19-s + 0.436·21-s − 0.208·23-s + 4/5·25-s + 0.769·27-s − 0.928·29-s + 0.538·31-s − 0.507·35-s − 0.986·37-s + 0.320·39-s − 0.312·41-s − 0.152·43-s + 0.447·45-s − 0.437·47-s + 1/7·49-s + 0.560·51-s + 1.51·53-s + 1.32·57-s + 1.04·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−9.203180653575062654359672426254, −8.529645336781118180497881820953, −7.14841316326545066045886217042, −6.43339660293291446952337401723, −5.85005995324875418186457648347, −5.20967988271522414086153489032, −4.20796727851533691291580413215, −2.72085251260713353689009977090, −1.66748781498573342728564782283, 0,
1.66748781498573342728564782283, 2.72085251260713353689009977090, 4.20796727851533691291580413215, 5.20967988271522414086153489032, 5.85005995324875418186457648347, 6.43339660293291446952337401723, 7.14841316326545066045886217042, 8.529645336781118180497881820953, 9.203180653575062654359672426254
