L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 4·11-s − 2·13-s − 15-s + 2·17-s − 4·19-s + 21-s + 25-s − 27-s − 10·29-s + 4·33-s − 35-s + 6·37-s + 2·39-s − 6·41-s − 4·43-s + 45-s − 8·47-s + 49-s − 2·51-s + 6·53-s − 4·55-s + 4·57-s − 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.696·33-s − 0.169·35-s + 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.149·45-s − 1.16·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.539·55-s + 0.529·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 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 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.975450018667804597038316719251, −9.082259719636065240272815793133, −7.955913714504644673844453268748, −7.18329359631779929148111937833, −6.14158704727116024821617582822, −5.44060214897147557533069905801, −4.53435327548118041491444384735, −3.15914258096067914389137866388, −1.93913660460555289279452591140, 0,
1.93913660460555289279452591140, 3.15914258096067914389137866388, 4.53435327548118041491444384735, 5.44060214897147557533069905801, 6.14158704727116024821617582822, 7.18329359631779929148111937833, 7.955913714504644673844453268748, 9.082259719636065240272815793133, 9.975450018667804597038316719251