| L(s) = 1 | − 2·3-s + 7-s + 9-s − 5·11-s + 13-s − 4·17-s + 7·19-s − 2·21-s − 23-s + 4·27-s + 5·29-s + 2·31-s + 10·33-s − 2·37-s − 2·39-s + 11·41-s + 43-s − 8·47-s − 6·49-s + 8·51-s − 14·57-s − 14·59-s + 10·61-s + 63-s − 8·67-s + 2·69-s − 10·71-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.50·11-s + 0.277·13-s − 0.970·17-s + 1.60·19-s − 0.436·21-s − 0.208·23-s + 0.769·27-s + 0.928·29-s + 0.359·31-s + 1.74·33-s − 0.328·37-s − 0.320·39-s + 1.71·41-s + 0.152·43-s − 1.16·47-s − 6/7·49-s + 1.12·51-s − 1.85·57-s − 1.82·59-s + 1.28·61-s + 0.125·63-s − 0.977·67-s + 0.240·69-s − 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.86801455550478310692678697649, −7.24471645893977572054164471045, −6.30042580462697723697740607443, −5.77470394317946026301406730981, −4.94164769941009819657454798099, −4.64590344806883541112647457240, −3.26611037528121319392221485408, −2.44145620499863655877298259822, −1.12842191488105739380512446952, 0,
1.12842191488105739380512446952, 2.44145620499863655877298259822, 3.26611037528121319392221485408, 4.64590344806883541112647457240, 4.94164769941009819657454798099, 5.77470394317946026301406730981, 6.30042580462697723697740607443, 7.24471645893977572054164471045, 7.86801455550478310692678697649
