L(s) = 1 | + 3-s + 5-s − 7-s − 2·9-s + 3·13-s + 15-s − 7·17-s + 19-s − 21-s − 5·23-s + 25-s − 5·27-s + 5·29-s + 10·31-s − 35-s − 2·37-s + 3·39-s + 2·41-s − 6·43-s − 2·45-s − 6·49-s − 7·51-s − 9·53-s + 57-s + 7·59-s + 4·61-s + 2·63-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.832·13-s + 0.258·15-s − 1.69·17-s + 0.229·19-s − 0.218·21-s − 1.04·23-s + 1/5·25-s − 0.962·27-s + 0.928·29-s + 1.79·31-s − 0.169·35-s − 0.328·37-s + 0.480·39-s + 0.312·41-s − 0.914·43-s − 0.298·45-s − 6/7·49-s − 0.980·51-s − 1.23·53-s + 0.132·57-s + 0.911·59-s + 0.512·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.992163571176218018110340599906, −6.80014773191940009506494915973, −6.37866299170461284822481763215, −5.72906766467486382197758944621, −4.72673976763971238769726906697, −4.00540344967317129351925958514, −3.03876626292382033847060855430, −2.48206108209228295528698972398, −1.47693515273990449433209054058, 0,
1.47693515273990449433209054058, 2.48206108209228295528698972398, 3.03876626292382033847060855430, 4.00540344967317129351925958514, 4.72673976763971238769726906697, 5.72906766467486382197758944621, 6.37866299170461284822481763215, 6.80014773191940009506494915973, 7.992163571176218018110340599906