| L(s) = 1 | − 3-s − 2·5-s + 9-s − 4·11-s + 13-s + 2·15-s + 17-s − 4·19-s − 25-s − 27-s − 6·29-s − 8·31-s + 4·33-s − 10·37-s − 39-s − 10·41-s − 2·45-s − 4·47-s − 7·49-s − 51-s − 2·53-s + 8·55-s + 4·57-s − 4·59-s − 6·61-s − 2·65-s + 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 0.516·15-s + 0.242·17-s − 0.917·19-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s + 0.696·33-s − 1.64·37-s − 0.160·39-s − 1.56·41-s − 0.298·45-s − 0.583·47-s − 49-s − 0.140·51-s − 0.274·53-s + 1.07·55-s + 0.529·57-s − 0.520·59-s − 0.768·61-s − 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.08926248419773, −15.56427189718393, −15.20496507065658, −14.70142004376352, −13.84765846145665, −13.34277007265836, −12.67417154584484, −12.43167992745354, −11.68482692859479, −11.14378998697203, −10.75755156157660, −10.22906403480160, −9.532408790851345, −8.796944082597492, −8.151919962781488, −7.735459088761790, −7.109682013559102, −6.507737172199115, −5.728466915049854, −5.150513420459535, −4.666628641438937, −3.609042437640877, −3.490555138765390, −2.235352240501869, −1.549481065690417, 0, 0,
1.549481065690417, 2.235352240501869, 3.490555138765390, 3.609042437640877, 4.666628641438937, 5.150513420459535, 5.728466915049854, 6.507737172199115, 7.109682013559102, 7.735459088761790, 8.151919962781488, 8.796944082597492, 9.532408790851345, 10.22906403480160, 10.75755156157660, 11.14378998697203, 11.68482692859479, 12.43167992745354, 12.67417154584484, 13.34277007265836, 13.84765846145665, 14.70142004376352, 15.20496507065658, 15.56427189718393, 16.08926248419773
