| L(s) = 1 | − 5-s + 7-s − 3·9-s + 2·13-s + 2·17-s − 8·19-s + 8·23-s + 25-s − 6·29-s − 35-s + 2·37-s − 6·41-s − 8·43-s + 3·45-s − 8·47-s + 49-s + 2·53-s − 8·59-s + 2·61-s − 3·63-s − 2·65-s − 8·67-s + 10·73-s − 16·79-s + 9·81-s − 16·83-s − 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 9-s + 0.554·13-s + 0.485·17-s − 1.83·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s − 0.169·35-s + 0.328·37-s − 0.937·41-s − 1.21·43-s + 0.447·45-s − 1.16·47-s + 1/7·49-s + 0.274·53-s − 1.04·59-s + 0.256·61-s − 0.377·63-s − 0.248·65-s − 0.977·67-s + 1.17·73-s − 1.80·79-s + 81-s − 1.75·83-s − 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.542888456146907943310150762194, −8.101896361223496279215404471520, −7.08238801180409597632017840133, −6.32328650531859522942267529605, −5.44974046246333672373644401925, −4.65338050774040712735752991165, −3.65431614032536343581686192220, −2.82369671155409779598436735706, −1.57106449768643065940389037433, 0,
1.57106449768643065940389037433, 2.82369671155409779598436735706, 3.65431614032536343581686192220, 4.65338050774040712735752991165, 5.44974046246333672373644401925, 6.32328650531859522942267529605, 7.08238801180409597632017840133, 8.101896361223496279215404471520, 8.542888456146907943310150762194
