| L(s) = 1 | + 2·5-s − 2·7-s − 2·13-s + 2·17-s − 4·19-s + 6·23-s − 25-s − 2·29-s − 4·35-s − 2·37-s − 6·41-s + 8·43-s + 2·47-s − 3·49-s + 10·53-s − 4·59-s − 14·61-s − 4·65-s − 4·67-s + 6·71-s − 8·73-s + 2·79-s + 12·83-s + 4·85-s − 8·89-s + 4·91-s − 8·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 0.554·13-s + 0.485·17-s − 0.917·19-s + 1.25·23-s − 1/5·25-s − 0.371·29-s − 0.676·35-s − 0.328·37-s − 0.937·41-s + 1.21·43-s + 0.291·47-s − 3/7·49-s + 1.37·53-s − 0.520·59-s − 1.79·61-s − 0.496·65-s − 0.488·67-s + 0.712·71-s − 0.936·73-s + 0.225·79-s + 1.31·83-s + 0.433·85-s − 0.847·89-s + 0.419·91-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.30842282135600389328358638669, −6.68980127089374670017722942183, −6.04988360156779548139432602876, −5.45078867161489944015224292368, −4.71233473932615722787572101574, −3.80403844890612624657534749589, −2.95882525028797804453722680623, −2.27522385118068300645993558567, −1.31515308864213113911805720827, 0,
1.31515308864213113911805720827, 2.27522385118068300645993558567, 2.95882525028797804453722680623, 3.80403844890612624657534749589, 4.71233473932615722787572101574, 5.45078867161489944015224292368, 6.04988360156779548139432602876, 6.68980127089374670017722942183, 7.30842282135600389328358638669
