| L(s) = 1 | + 3-s − 5·7-s + 9-s + 6·11-s + 3·13-s + 19-s − 5·21-s − 2·23-s + 27-s − 6·29-s − 3·31-s + 6·33-s − 6·37-s + 3·39-s − 4·41-s − 11·43-s + 10·47-s + 18·49-s + 8·53-s + 57-s − 6·59-s − 3·61-s − 5·63-s + 67-s − 2·69-s + 12·71-s + 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.88·7-s + 1/3·9-s + 1.80·11-s + 0.832·13-s + 0.229·19-s − 1.09·21-s − 0.417·23-s + 0.192·27-s − 1.11·29-s − 0.538·31-s + 1.04·33-s − 0.986·37-s + 0.480·39-s − 0.624·41-s − 1.67·43-s + 1.45·47-s + 18/7·49-s + 1.09·53-s + 0.132·57-s − 0.781·59-s − 0.384·61-s − 0.629·63-s + 0.122·67-s − 0.240·69-s + 1.42·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.54673828853044, −12.98856064315882, −12.50306704818574, −12.10356158974483, −11.71718334639369, −10.98816968010061, −10.56690237547178, −9.909893664824324, −9.498637433850881, −9.286451883402019, −8.704485923013486, −8.402847150155020, −7.522151322721920, −7.026670387836763, −6.609619524621433, −6.325807173981753, −5.695524579070230, −5.166555747951912, −4.076825980407014, −3.835531401066661, −3.508246046024709, −2.993178811905221, −2.152942209724067, −1.572325711275949, −0.8500857866682834, 0,
0.8500857866682834, 1.572325711275949, 2.152942209724067, 2.993178811905221, 3.508246046024709, 3.835531401066661, 4.076825980407014, 5.166555747951912, 5.695524579070230, 6.325807173981753, 6.609619524621433, 7.026670387836763, 7.522151322721920, 8.402847150155020, 8.704485923013486, 9.286451883402019, 9.498637433850881, 9.909893664824324, 10.56690237547178, 10.98816968010061, 11.71718334639369, 12.10356158974483, 12.50306704818574, 12.98856064315882, 13.54673828853044
