| L(s) = 1 | − 60·3-s − 4.34e3·7-s − 1.60e4·9-s − 9.36e4·11-s + 1.22e4·13-s + 3.19e5·17-s + 5.53e5·19-s + 2.60e5·21-s − 7.12e5·23-s + 2.14e6·27-s + 2.07e6·29-s + 6.42e6·31-s + 5.61e6·33-s + 1.81e7·37-s − 7.34e5·39-s + 9.03e6·41-s + 1.95e7·43-s − 1.84e7·47-s − 2.14e7·49-s − 1.91e7·51-s − 1.02e7·53-s − 3.32e7·57-s − 1.21e8·59-s − 4.59e7·61-s + 6.98e7·63-s + 5.05e7·67-s + 4.27e7·69-s + ⋯ |
| L(s) = 1 | − 0.427·3-s − 0.683·7-s − 0.817·9-s − 1.92·11-s + 0.118·13-s + 0.928·17-s + 0.974·19-s + 0.292·21-s − 0.531·23-s + 0.777·27-s + 0.545·29-s + 1.24·31-s + 0.824·33-s + 1.59·37-s − 0.0508·39-s + 0.499·41-s + 0.874·43-s − 0.552·47-s − 0.532·49-s − 0.396·51-s − 0.178·53-s − 0.416·57-s − 1.30·59-s − 0.424·61-s + 0.558·63-s + 0.306·67-s + 0.227·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 20 p T + p^{9} T^{2} \) |
| 7 | \( 1 + 4344 T + p^{9} T^{2} \) |
| 11 | \( 1 + 93644 T + p^{9} T^{2} \) |
| 13 | \( 1 - 12242 T + p^{9} T^{2} \) |
| 17 | \( 1 - 319598 T + p^{9} T^{2} \) |
| 19 | \( 1 - 553516 T + p^{9} T^{2} \) |
| 23 | \( 1 + 712936 T + p^{9} T^{2} \) |
| 29 | \( 1 - 2075838 T + p^{9} T^{2} \) |
| 31 | \( 1 - 6420448 T + p^{9} T^{2} \) |
| 37 | \( 1 - 18197754 T + p^{9} T^{2} \) |
| 41 | \( 1 - 9033834 T + p^{9} T^{2} \) |
| 43 | \( 1 - 19594732 T + p^{9} T^{2} \) |
| 47 | \( 1 + 18484176 T + p^{9} T^{2} \) |
| 53 | \( 1 + 10255766 T + p^{9} T^{2} \) |
| 59 | \( 1 + 121666556 T + p^{9} T^{2} \) |
| 61 | \( 1 + 45948962 T + p^{9} T^{2} \) |
| 67 | \( 1 - 50535428 T + p^{9} T^{2} \) |
| 71 | \( 1 + 267044680 T + p^{9} T^{2} \) |
| 73 | \( 1 - 176213366 T + p^{9} T^{2} \) |
| 79 | \( 1 - 269685680 T + p^{9} T^{2} \) |
| 83 | \( 1 + 2735332 p T + p^{9} T^{2} \) |
| 89 | \( 1 - 72141594 T + p^{9} T^{2} \) |
| 97 | \( 1 + 228776546 T + p^{9} T^{2} \) |
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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
−9.504158899055394853288638541792, −8.172041614096452760690485361313, −7.64165625775996329951324001058, −6.25117632419758474142930558428, −5.61160636542449318769422346769, −4.71079964987999635880306594819, −3.14879022623480751603654328610, −2.61684822608904123142848443193, −0.901342663691866464658693680048, 0,
0.901342663691866464658693680048, 2.61684822608904123142848443193, 3.14879022623480751603654328610, 4.71079964987999635880306594819, 5.61160636542449318769422346769, 6.25117632419758474142930558428, 7.64165625775996329951324001058, 8.172041614096452760690485361313, 9.504158899055394853288638541792
