| L(s) = 1 | + 5-s + 11-s + 2·13-s − 3·17-s + 5·19-s + 3·23-s − 4·25-s + 6·29-s − 31-s − 5·37-s + 10·41-s − 4·43-s − 47-s + 9·53-s + 55-s − 3·59-s + 3·61-s + 2·65-s + 11·67-s − 16·71-s + 7·73-s − 11·79-s + 4·83-s − 3·85-s + 9·89-s + 5·95-s + 6·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.301·11-s + 0.554·13-s − 0.727·17-s + 1.14·19-s + 0.625·23-s − 4/5·25-s + 1.11·29-s − 0.179·31-s − 0.821·37-s + 1.56·41-s − 0.609·43-s − 0.145·47-s + 1.23·53-s + 0.134·55-s − 0.390·59-s + 0.384·61-s + 0.248·65-s + 1.34·67-s − 1.89·71-s + 0.819·73-s − 1.23·79-s + 0.439·83-s − 0.325·85-s + 0.953·89-s + 0.512·95-s + 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.198776312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.198776312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−8.733722664677072743310929907108, −7.80706592603852065568869669597, −7.05053906005695924176616820982, −6.32062198333587856968601571432, −5.61098230910717143267119759444, −4.80503447589492919600099799779, −3.89222642104121663412460242149, −3.00697938382676860276621951385, −1.98881442880812381861732336932, −0.905675788675921911783531337807,
0.905675788675921911783531337807, 1.98881442880812381861732336932, 3.00697938382676860276621951385, 3.89222642104121663412460242149, 4.80503447589492919600099799779, 5.61098230910717143267119759444, 6.32062198333587856968601571432, 7.05053906005695924176616820982, 7.80706592603852065568869669597, 8.733722664677072743310929907108
