| L(s) = 1 | + 5-s − 2·11-s − 4·13-s + 2·17-s + 2·19-s − 4·23-s + 25-s + 2·29-s + 6·31-s − 6·37-s + 6·41-s − 4·43-s − 8·53-s − 2·55-s + 10·61-s − 4·65-s − 12·67-s + 14·71-s − 4·73-s − 8·79-s + 12·83-s + 2·85-s − 14·89-s + 2·95-s − 8·97-s − 6·101-s + 8·103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.603·11-s − 1.10·13-s + 0.485·17-s + 0.458·19-s − 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.07·31-s − 0.986·37-s + 0.937·41-s − 0.609·43-s − 1.09·53-s − 0.269·55-s + 1.28·61-s − 0.496·65-s − 1.46·67-s + 1.66·71-s − 0.468·73-s − 0.900·79-s + 1.31·83-s + 0.216·85-s − 1.48·89-s + 0.205·95-s − 0.812·97-s − 0.597·101-s + 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−7.49378382849519338551139555729, −6.71777086311712217667538867430, −6.02950541898577638443177295578, −5.25004629115447600927052627944, −4.80911851160011857197385375307, −3.85136711546810483700264600836, −2.89614729467649845923785053891, −2.31926220702263285836837162077, −1.27648647987674642212559034199, 0,
1.27648647987674642212559034199, 2.31926220702263285836837162077, 2.89614729467649845923785053891, 3.85136711546810483700264600836, 4.80911851160011857197385375307, 5.25004629115447600927052627944, 6.02950541898577638443177295578, 6.71777086311712217667538867430, 7.49378382849519338551139555729
