| L(s) = 1 | + 2·5-s − 7-s − 3·9-s − 4·11-s − 13-s − 2·17-s − 4·19-s + 23-s − 25-s + 2·29-s − 2·35-s − 2·37-s − 6·41-s − 4·43-s − 6·45-s − 8·47-s + 49-s − 6·53-s − 8·55-s + 12·59-s − 10·61-s + 3·63-s − 2·65-s + 4·67-s − 12·71-s + 10·73-s + 4·77-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 9-s − 1.20·11-s − 0.277·13-s − 0.485·17-s − 0.917·19-s + 0.208·23-s − 1/5·25-s + 0.371·29-s − 0.338·35-s − 0.328·37-s − 0.937·41-s − 0.609·43-s − 0.894·45-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s + 1.56·59-s − 1.28·61-s + 0.377·63-s − 0.248·65-s + 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.79864550111422, −13.42144977456433, −13.08134916144849, −12.65324389244134, −11.97515647271154, −11.57394916567163, −10.94143683321912, −10.48331251805409, −10.18352314336550, −9.547478580884993, −9.179069211618781, −8.458765274579223, −8.251039353859236, −7.633250576273850, −6.877040894567898, −6.435861698055513, −6.043251013534512, −5.346045034767667, −5.103230341365897, −4.470691530443188, −3.607599906523819, −3.068443793708690, −2.444405269269463, −2.133587924235460, −1.310327524541878, 0, 0,
1.310327524541878, 2.133587924235460, 2.444405269269463, 3.068443793708690, 3.607599906523819, 4.470691530443188, 5.103230341365897, 5.346045034767667, 6.043251013534512, 6.435861698055513, 6.877040894567898, 7.633250576273850, 8.251039353859236, 8.458765274579223, 9.179069211618781, 9.547478580884993, 10.18352314336550, 10.48331251805409, 10.94143683321912, 11.57394916567163, 11.97515647271154, 12.65324389244134, 13.08134916144849, 13.42144977456433, 13.79864550111422
