| L(s) = 1 | + 3-s + 9-s + 5·11-s + 2·13-s − 6·17-s + 2·19-s + 5·23-s + 27-s − 5·29-s + 4·31-s + 5·33-s − 37-s + 2·39-s − 12·41-s + 5·43-s + 2·47-s − 6·51-s − 14·53-s + 2·57-s − 2·59-s − 5·67-s + 5·69-s + 9·71-s − 10·73-s − 11·79-s + 81-s + 16·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.50·11-s + 0.554·13-s − 1.45·17-s + 0.458·19-s + 1.04·23-s + 0.192·27-s − 0.928·29-s + 0.718·31-s + 0.870·33-s − 0.164·37-s + 0.320·39-s − 1.87·41-s + 0.762·43-s + 0.291·47-s − 0.840·51-s − 1.92·53-s + 0.264·57-s − 0.260·59-s − 0.610·67-s + 0.601·69-s + 1.06·71-s − 1.17·73-s − 1.23·79-s + 1/9·81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.75631007645254, −13.91512592663789, −13.69622833747508, −13.21277236894008, −12.58883340056497, −12.10192800537565, −11.32899936706326, −11.24992226394051, −10.53834207930342, −9.831807459564469, −9.289060530901414, −8.953707755383601, −8.552708800604202, −7.879085643846099, −7.208980096653979, −6.651672344714214, −6.406987851917168, −5.584154951625952, −4.820882810305746, −4.310751004855653, −3.716210821162718, −3.189418968733054, −2.467579277180440, −1.600146291538986, −1.214073551645340, 0,
1.214073551645340, 1.600146291538986, 2.467579277180440, 3.189418968733054, 3.716210821162718, 4.310751004855653, 4.820882810305746, 5.584154951625952, 6.406987851917168, 6.651672344714214, 7.208980096653979, 7.879085643846099, 8.552708800604202, 8.953707755383601, 9.289060530901414, 9.831807459564469, 10.53834207930342, 11.24992226394051, 11.32899936706326, 12.10192800537565, 12.58883340056497, 13.21277236894008, 13.69622833747508, 13.91512592663789, 14.75631007645254
