L(s) = 1 | + 5-s − 4·7-s − 3·9-s − 4·11-s − 2·13-s − 2·17-s − 4·19-s + 4·23-s + 25-s + 8·31-s − 4·35-s − 6·37-s + 6·41-s + 8·43-s − 3·45-s − 4·47-s + 9·49-s + 6·53-s − 4·55-s − 4·59-s + 2·61-s + 12·63-s − 2·65-s + 8·67-s + 6·73-s + 16·77-s + 9·81-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.51·7-s − 9-s − 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 1.43·31-s − 0.676·35-s − 0.986·37-s + 0.937·41-s + 1.21·43-s − 0.447·45-s − 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.539·55-s − 0.520·59-s + 0.256·61-s + 1.51·63-s − 0.248·65-s + 0.977·67-s + 0.702·73-s + 1.82·77-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−15.37549537310899, −14.74048752760684, −14.12860792490237, −13.60331584853034, −13.15571969785026, −12.62316671210797, −12.37864159403511, −11.47539359568889, −10.96791682530584, −10.29915579329998, −10.08156225923280, −9.293785194713515, −8.911913213139615, −8.318545880791698, −7.646250991061884, −6.928786853465612, −6.430432123091344, −5.940951588519136, −5.322372758807355, −4.747384525950520, −3.904337127808482, −3.046340851169198, −2.664694133115335, −2.181195691877715, −0.7322024083033291, 0,
0.7322024083033291, 2.181195691877715, 2.664694133115335, 3.046340851169198, 3.904337127808482, 4.747384525950520, 5.322372758807355, 5.940951588519136, 6.430432123091344, 6.928786853465612, 7.646250991061884, 8.318545880791698, 8.911913213139615, 9.293785194713515, 10.08156225923280, 10.29915579329998, 10.96791682530584, 11.47539359568889, 12.37864159403511, 12.62316671210797, 13.15571969785026, 13.60331584853034, 14.12860792490237, 14.74048752760684, 15.37549537310899