L(s) = 1 | + 4·5-s + 7-s − 4·11-s − 13-s + 3·17-s − 2·19-s + 3·23-s + 11·25-s − 9·29-s + 31-s + 4·35-s + 10·37-s + 10·41-s + 43-s + 10·47-s + 49-s + 9·53-s − 16·55-s + 3·59-s − 6·61-s − 4·65-s + 13·67-s + 13·71-s + 10·73-s − 4·77-s − 10·79-s − 4·83-s + ⋯ |
L(s) = 1 | + 1.78·5-s + 0.377·7-s − 1.20·11-s − 0.277·13-s + 0.727·17-s − 0.458·19-s + 0.625·23-s + 11/5·25-s − 1.67·29-s + 0.179·31-s + 0.676·35-s + 1.64·37-s + 1.56·41-s + 0.152·43-s + 1.45·47-s + 1/7·49-s + 1.23·53-s − 2.15·55-s + 0.390·59-s − 0.768·61-s − 0.496·65-s + 1.58·67-s + 1.54·71-s + 1.17·73-s − 0.455·77-s − 1.12·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.668567521\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.668567521\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.885063319772254523354507623868, −7.915431568080165390573593813744, −7.27578047424983475531404009371, −6.27119751911351423248379961190, −5.51590591684487829752924036873, −5.24616955024848367729689849260, −4.08214928041124711322444578919, −2.64961138107861303786018429715, −2.26447710052247839812854984658, −1.03751248798011982769049144291,
1.03751248798011982769049144291, 2.26447710052247839812854984658, 2.64961138107861303786018429715, 4.08214928041124711322444578919, 5.24616955024848367729689849260, 5.51590591684487829752924036873, 6.27119751911351423248379961190, 7.27578047424983475531404009371, 7.915431568080165390573593813744, 8.885063319772254523354507623868