| L(s) = 1 | − 5-s − 3·9-s − 4·11-s + 2·13-s − 2·17-s + 4·19-s − 4·23-s + 25-s − 2·29-s − 8·31-s + 6·37-s + 6·41-s + 8·43-s + 3·45-s + 4·47-s + 6·53-s + 4·55-s − 4·59-s + 2·61-s − 2·65-s − 8·67-s + 6·73-s + 9·81-s − 16·83-s + 2·85-s + 6·89-s − 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 9-s − 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.986·37-s + 0.937·41-s + 1.21·43-s + 0.447·45-s + 0.583·47-s + 0.824·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s − 0.248·65-s − 0.977·67-s + 0.702·73-s + 81-s − 1.75·83-s + 0.216·85-s + 0.635·89-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.122110356\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.122110356\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.454267864352545795806495542150, −7.67375816764907867780022374833, −7.28236915503800031844503481655, −5.93296579879359794523720946560, −5.71297380596599560620236986392, −4.69948739897509711013983659515, −3.79782147759071587315933837414, −2.95971176692665828505622566750, −2.14273353770490399369013857078, −0.58317500635071871496421307783,
0.58317500635071871496421307783, 2.14273353770490399369013857078, 2.95971176692665828505622566750, 3.79782147759071587315933837414, 4.69948739897509711013983659515, 5.71297380596599560620236986392, 5.93296579879359794523720946560, 7.28236915503800031844503481655, 7.67375816764907867780022374833, 8.454267864352545795806495542150
