| L(s) = 1 | + 3-s − 2·5-s + 9-s − 2·15-s + 8·19-s − 16·23-s + 3·25-s + 27-s − 12·29-s − 8·43-s − 2·45-s + 16·47-s + 2·49-s + 20·53-s + 8·57-s − 8·67-s − 16·69-s − 28·73-s + 3·75-s + 81-s − 12·87-s − 16·95-s + 4·97-s − 28·101-s + 32·115-s − 22·121-s − 4·125-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.516·15-s + 1.83·19-s − 3.33·23-s + 3/5·25-s + 0.192·27-s − 2.22·29-s − 1.21·43-s − 0.298·45-s + 2.33·47-s + 2/7·49-s + 2.74·53-s + 1.05·57-s − 0.977·67-s − 1.92·69-s − 3.27·73-s + 0.346·75-s + 1/9·81-s − 1.28·87-s − 1.64·95-s + 0.406·97-s − 2.78·101-s + 2.98·115-s − 2·121-s − 0.357·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.454184844352058513106035895572, −7.929044973464791963060046214031, −7.61156995905808074770369477086, −7.34594365176835478558310775458, −6.87444493455441081760547968335, −5.99885962909318068324510418820, −5.63560405748275108473435534610, −5.28897888504504330389281711769, −4.13728859524865050162716547657, −4.11858864877398701222065258089, −3.62493705637031726202792867085, −2.86501930332949571186241415418, −2.19180563382997996757520036314, −1.38232553241565659733165199365, 0,
1.38232553241565659733165199365, 2.19180563382997996757520036314, 2.86501930332949571186241415418, 3.62493705637031726202792867085, 4.11858864877398701222065258089, 4.13728859524865050162716547657, 5.28897888504504330389281711769, 5.63560405748275108473435534610, 5.99885962909318068324510418820, 6.87444493455441081760547968335, 7.34594365176835478558310775458, 7.61156995905808074770369477086, 7.929044973464791963060046214031, 8.454184844352058513106035895572
