| L(s) = 1 | + 3-s + 5-s + 9-s + 2·13-s + 15-s − 2·17-s + 4·19-s + 25-s + 27-s + 2·29-s − 10·37-s + 2·39-s − 10·41-s + 4·43-s + 45-s − 8·47-s − 7·49-s − 2·51-s − 10·53-s + 4·57-s + 4·59-s + 2·61-s + 2·65-s − 12·67-s + 8·71-s − 10·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.554·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 1.64·37-s + 0.320·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s − 1.16·47-s − 49-s − 0.280·51-s − 1.37·53-s + 0.529·57-s + 0.520·59-s + 0.256·61-s + 0.248·65-s − 1.46·67-s + 0.949·71-s − 1.17·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.45321514062981, −14.88476678722055, −14.25108989409567, −13.90495202074318, −13.35928539951467, −12.98758520531127, −12.26515854730900, −11.73385704390607, −11.14858786303303, −10.48541506539559, −10.03920193380913, −9.429311114853201, −8.933614044804268, −8.385135639329550, −7.869765249767162, −7.129221463097211, −6.594977055346115, −6.071097909454496, −5.149555251941368, −4.861582671100601, −3.865874751857574, −3.323959007578207, −2.720844950966845, −1.783181842092050, −1.312943511969394, 0,
1.312943511969394, 1.783181842092050, 2.720844950966845, 3.323959007578207, 3.865874751857574, 4.861582671100601, 5.149555251941368, 6.071097909454496, 6.594977055346115, 7.129221463097211, 7.869765249767162, 8.385135639329550, 8.933614044804268, 9.429311114853201, 10.03920193380913, 10.48541506539559, 11.14858786303303, 11.73385704390607, 12.26515854730900, 12.98758520531127, 13.35928539951467, 13.90495202074318, 14.25108989409567, 14.88476678722055, 15.45321514062981
