| L(s) = 1 | + 5-s + 7-s + 13-s − 17-s − 4·19-s − 8·23-s − 4·25-s − 8·29-s + 35-s + 7·37-s − 8·41-s + 11·43-s − 47-s − 6·49-s − 2·53-s + 14·59-s + 8·61-s + 65-s − 8·67-s + 9·71-s + 4·73-s + 2·79-s − 85-s + 4·89-s + 91-s − 4·95-s + 8·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.277·13-s − 0.242·17-s − 0.917·19-s − 1.66·23-s − 4/5·25-s − 1.48·29-s + 0.169·35-s + 1.15·37-s − 1.24·41-s + 1.67·43-s − 0.145·47-s − 6/7·49-s − 0.274·53-s + 1.82·59-s + 1.02·61-s + 0.124·65-s − 0.977·67-s + 1.06·71-s + 0.468·73-s + 0.225·79-s − 0.108·85-s + 0.423·89-s + 0.104·91-s − 0.410·95-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.17098518765178, −12.75519316863724, −12.34674127214759, −11.57717894421839, −11.39847998876301, −10.96470768342082, −10.20746594067268, −10.01543994628227, −9.498217262590306, −8.917965741466379, −8.511772633190076, −7.886507154471373, −7.681550076385156, −6.958802318300737, −6.380684859146945, −5.921320361507268, −5.659661732959502, −4.896013067282820, −4.416026373162720, −3.766594087372089, −3.551538351357170, −2.420480939417923, −2.136002470696981, −1.684191688323479, −0.7668307246592781, 0,
0.7668307246592781, 1.684191688323479, 2.136002470696981, 2.420480939417923, 3.551538351357170, 3.766594087372089, 4.416026373162720, 4.896013067282820, 5.659661732959502, 5.921320361507268, 6.380684859146945, 6.958802318300737, 7.681550076385156, 7.886507154471373, 8.511772633190076, 8.917965741466379, 9.498217262590306, 10.01543994628227, 10.20746594067268, 10.96470768342082, 11.39847998876301, 11.57717894421839, 12.34674127214759, 12.75519316863724, 13.17098518765178
