| L(s) = 1 | + 5-s − 2·7-s − 4·11-s − 13-s + 4·17-s − 6·19-s + 25-s + 4·29-s + 10·31-s − 2·35-s − 2·37-s + 6·41-s + 8·43-s − 8·47-s − 3·49-s + 4·53-s − 4·55-s + 12·59-s + 2·61-s − 65-s + 10·67-s − 6·73-s + 8·77-s − 12·79-s − 4·83-s + 4·85-s − 14·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.20·11-s − 0.277·13-s + 0.970·17-s − 1.37·19-s + 1/5·25-s + 0.742·29-s + 1.79·31-s − 0.338·35-s − 0.328·37-s + 0.937·41-s + 1.21·43-s − 1.16·47-s − 3/7·49-s + 0.549·53-s − 0.539·55-s + 1.56·59-s + 0.256·61-s − 0.124·65-s + 1.22·67-s − 0.702·73-s + 0.911·77-s − 1.35·79-s − 0.439·83-s + 0.433·85-s − 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−7.29447012753942569603024574429, −6.63788780916148030388537766699, −6.01034373630005184743069112387, −5.38102803778657028142426061955, −4.63041394126600343780500901084, −3.83505574553953599860126815787, −2.72011942988615649765537273343, −2.53360959103320458637797496723, −1.16379173746708222067006623856, 0,
1.16379173746708222067006623856, 2.53360959103320458637797496723, 2.72011942988615649765537273343, 3.83505574553953599860126815787, 4.63041394126600343780500901084, 5.38102803778657028142426061955, 6.01034373630005184743069112387, 6.63788780916148030388537766699, 7.29447012753942569603024574429
