L(s) = 1 | − 5-s − 4·7-s + 6·13-s + 2·17-s + 4·19-s − 8·23-s + 25-s − 6·29-s + 4·35-s + 6·37-s − 10·41-s − 4·43-s + 8·47-s + 9·49-s + 10·53-s − 6·61-s − 6·65-s − 4·67-s − 14·73-s − 16·79-s − 12·83-s − 2·85-s − 2·89-s − 24·91-s − 4·95-s + 2·97-s − 14·101-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.51·7-s + 1.66·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 1.11·29-s + 0.676·35-s + 0.986·37-s − 1.56·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s + 1.37·53-s − 0.768·61-s − 0.744·65-s − 0.488·67-s − 1.63·73-s − 1.80·79-s − 1.31·83-s − 0.216·85-s − 0.211·89-s − 2.51·91-s − 0.410·95-s + 0.203·97-s − 1.39·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.454184844352058513106035895572, −7.61156995905808074770369477086, −6.87444493455441081760547968335, −5.99885962909318068324510418820, −5.63560405748275108473435534610, −4.11858864877398701222065258089, −3.62493705637031726202792867085, −2.86501930332949571186241415418, −1.38232553241565659733165199365, 0,
1.38232553241565659733165199365, 2.86501930332949571186241415418, 3.62493705637031726202792867085, 4.11858864877398701222065258089, 5.63560405748275108473435534610, 5.99885962909318068324510418820, 6.87444493455441081760547968335, 7.61156995905808074770369477086, 8.454184844352058513106035895572