| L(s) = 1 | − 3-s − 5-s − 7-s − 2·9-s + 11-s − 4·13-s + 15-s − 6·17-s − 2·19-s + 21-s + 23-s − 4·25-s + 5·27-s + 2·29-s − 31-s − 33-s + 35-s − 9·37-s + 4·39-s + 6·41-s + 8·43-s + 2·45-s − 8·47-s + 49-s + 6·51-s + 10·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.301·11-s − 1.10·13-s + 0.258·15-s − 1.45·17-s − 0.458·19-s + 0.218·21-s + 0.208·23-s − 4/5·25-s + 0.962·27-s + 0.371·29-s − 0.179·31-s − 0.174·33-s + 0.169·35-s − 1.47·37-s + 0.640·39-s + 0.937·41-s + 1.21·43-s + 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.840·51-s + 1.37·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−11.32302259604564208279169259600, −10.47059015029117792874482226264, −9.329473956457167641788706515594, −8.441807132405645212577785816781, −7.20044051165186333263902429412, −6.33749430761827790522533904144, −5.18672240750960531371029490123, −4.05589871533530008614715770615, −2.50570574661996799886651591918, 0,
2.50570574661996799886651591918, 4.05589871533530008614715770615, 5.18672240750960531371029490123, 6.33749430761827790522533904144, 7.20044051165186333263902429412, 8.441807132405645212577785816781, 9.329473956457167641788706515594, 10.47059015029117792874482226264, 11.32302259604564208279169259600
