| L(s) = 1 | − 2·5-s + 4·11-s + 2·13-s − 2·17-s + 4·19-s + 8·23-s − 25-s + 6·29-s + 8·31-s − 6·37-s + 6·41-s − 4·43-s − 7·49-s − 2·53-s − 8·55-s + 4·59-s + 2·61-s − 4·65-s + 4·67-s − 8·71-s + 10·73-s − 8·79-s − 4·83-s + 4·85-s + 6·89-s − 8·95-s + 2·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s + 1.11·29-s + 1.43·31-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 49-s − 0.274·53-s − 1.07·55-s + 0.520·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s − 0.949·71-s + 1.17·73-s − 0.900·79-s − 0.439·83-s + 0.433·85-s + 0.635·89-s − 0.820·95-s + 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.376409401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.376409401\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−10.92388151188979644343896213523, −9.780091138821383013974264992430, −8.884556993307409273860919130264, −8.157389656224392853374347080198, −7.06857822018944724974086155331, −6.38250152569089720234352942221, −4.99175853408267534988469991920, −4.02227786472289347756607333391, −3.04433904291477058087427796742, −1.12835632773744922111480113749,
1.12835632773744922111480113749, 3.04433904291477058087427796742, 4.02227786472289347756607333391, 4.99175853408267534988469991920, 6.38250152569089720234352942221, 7.06857822018944724974086155331, 8.157389656224392853374347080198, 8.884556993307409273860919130264, 9.780091138821383013974264992430, 10.92388151188979644343896213523
