| L(s) = 1 | − 3·3-s + 5-s + 7-s + 6·9-s − 5·11-s − 5·13-s − 3·15-s − 7·17-s − 2·19-s − 3·21-s − 2·23-s + 25-s − 9·27-s + 7·29-s + 4·31-s + 15·33-s + 35-s − 6·37-s + 15·39-s − 12·41-s − 2·43-s + 6·45-s + 47-s + 49-s + 21·51-s − 5·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s − 1.50·11-s − 1.38·13-s − 0.774·15-s − 1.69·17-s − 0.458·19-s − 0.654·21-s − 0.417·23-s + 1/5·25-s − 1.73·27-s + 1.29·29-s + 0.718·31-s + 2.61·33-s + 0.169·35-s − 0.986·37-s + 2.40·39-s − 1.87·41-s − 0.304·43-s + 0.894·45-s + 0.145·47-s + 1/7·49-s + 2.94·51-s − 0.674·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−11.37548530993775479236655988699, −10.43970270366396486648555968518, −10.04608158910457177964610000230, −8.437696063431547440013268278023, −7.13289292005793470052045653622, −6.32900819422385960555488032719, −5.09619695718514137207067111766, −4.73560872492356455619359024880, −2.25342723502104844991389935790, 0,
2.25342723502104844991389935790, 4.73560872492356455619359024880, 5.09619695718514137207067111766, 6.32900819422385960555488032719, 7.13289292005793470052045653622, 8.437696063431547440013268278023, 10.04608158910457177964610000230, 10.43970270366396486648555968518, 11.37548530993775479236655988699
