| L(s) = 1 | + 3-s + 2·5-s + 9-s − 2·11-s + 2·15-s − 2·17-s − 2·23-s − 25-s + 27-s − 6·29-s − 4·31-s − 2·33-s − 6·37-s + 2·41-s + 2·45-s − 2·51-s + 6·53-s − 4·55-s − 12·59-s + 12·61-s − 12·67-s − 2·69-s + 10·71-s − 12·73-s − 75-s − 12·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.603·11-s + 0.516·15-s − 0.485·17-s − 0.417·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.348·33-s − 0.986·37-s + 0.312·41-s + 0.298·45-s − 0.280·51-s + 0.824·53-s − 0.539·55-s − 1.56·59-s + 1.53·61-s − 1.46·67-s − 0.240·69-s + 1.18·71-s − 1.40·73-s − 0.115·75-s − 1.35·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−7.38152233350264094134261020512, −6.77839586162877146619173499201, −5.82353022355995261190831502319, −5.49576071035680018496349915277, −4.53079591074299736281466223085, −3.78711488789932938503782185219, −2.91557507898183341940151924426, −2.14776468086628798905550068389, −1.54646372600141965125612720140, 0,
1.54646372600141965125612720140, 2.14776468086628798905550068389, 2.91557507898183341940151924426, 3.78711488789932938503782185219, 4.53079591074299736281466223085, 5.49576071035680018496349915277, 5.82353022355995261190831502319, 6.77839586162877146619173499201, 7.38152233350264094134261020512
