| L(s) = 1 | − 2·5-s − 7-s − 3·9-s − 4·11-s − 13-s − 6·17-s + 4·19-s − 8·23-s − 25-s − 6·29-s − 5·31-s + 2·35-s − 37-s + 7·41-s − 4·43-s + 6·45-s + 2·47-s − 6·49-s − 3·53-s + 8·55-s − 59-s − 5·61-s + 3·63-s + 2·65-s − 16·67-s − 5·71-s + 4·73-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s − 9-s − 1.20·11-s − 0.277·13-s − 1.45·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s − 1.11·29-s − 0.898·31-s + 0.338·35-s − 0.164·37-s + 1.09·41-s − 0.609·43-s + 0.894·45-s + 0.291·47-s − 6/7·49-s − 0.412·53-s + 1.07·55-s − 0.130·59-s − 0.640·61-s + 0.377·63-s + 0.248·65-s − 1.95·67-s − 0.593·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8732 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8732 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−7.36598226047801459827606941937, −6.27741975007207130961571044272, −5.74140012656830962570924648390, −4.99648492825474072284302828371, −4.18461191787707052668552001011, −3.45756818302031923476458846223, −2.70880558803766155326037781806, −1.91921861293205454547343691121, 0, 0,
1.91921861293205454547343691121, 2.70880558803766155326037781806, 3.45756818302031923476458846223, 4.18461191787707052668552001011, 4.99648492825474072284302828371, 5.74140012656830962570924648390, 6.27741975007207130961571044272, 7.36598226047801459827606941937
