| L(s) = 1 | − 2·5-s + 11-s − 2·13-s − 17-s − 8·19-s − 8·23-s − 25-s − 6·29-s − 8·31-s + 6·37-s − 2·41-s − 8·43-s − 4·47-s − 7·49-s − 6·53-s − 2·55-s − 8·59-s + 10·61-s + 4·65-s − 4·67-s − 10·73-s + 12·83-s + 2·85-s + 14·89-s + 16·95-s − 14·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.301·11-s − 0.554·13-s − 0.242·17-s − 1.83·19-s − 1.66·23-s − 1/5·25-s − 1.11·29-s − 1.43·31-s + 0.986·37-s − 0.312·41-s − 1.21·43-s − 0.583·47-s − 49-s − 0.824·53-s − 0.269·55-s − 1.04·59-s + 1.28·61-s + 0.496·65-s − 0.488·67-s − 1.17·73-s + 1.31·83-s + 0.216·85-s + 1.48·89-s + 1.64·95-s − 1.42·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−15.87955012115545, −15.02887765900210, −14.85569180645415, −14.48182147083148, −13.55721258031928, −13.12658311318383, −12.51367122730175, −12.09843954661148, −11.46451166520805, −11.07587641651604, −10.47412501154616, −9.778266110626677, −9.314473678031908, −8.592448746574907, −7.962465377364814, −7.749926322631088, −6.869852626693579, −6.392765943279498, −5.747360738628106, −4.952880785892710, −4.191625766484804, −3.931811322910601, −3.153689417239445, −2.118450181480476, −1.695366559687323, 0, 0,
1.695366559687323, 2.118450181480476, 3.153689417239445, 3.931811322910601, 4.191625766484804, 4.952880785892710, 5.747360738628106, 6.392765943279498, 6.869852626693579, 7.749926322631088, 7.962465377364814, 8.592448746574907, 9.314473678031908, 9.778266110626677, 10.47412501154616, 11.07587641651604, 11.46451166520805, 12.09843954661148, 12.51367122730175, 13.12658311318383, 13.55721258031928, 14.48182147083148, 14.85569180645415, 15.02887765900210, 15.87955012115545
