| L(s) = 1 | + 5-s + 3·7-s − 2·11-s − 2·13-s − 4·17-s + 8·19-s + 3·23-s + 25-s + 29-s + 3·35-s − 4·37-s − 5·41-s + 8·43-s + 7·47-s + 2·49-s + 2·53-s − 2·55-s − 14·59-s + 7·61-s − 2·65-s + 3·67-s + 2·71-s + 4·73-s − 6·77-s + 6·79-s + 9·83-s − 4·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s − 0.603·11-s − 0.554·13-s − 0.970·17-s + 1.83·19-s + 0.625·23-s + 1/5·25-s + 0.185·29-s + 0.507·35-s − 0.657·37-s − 0.780·41-s + 1.21·43-s + 1.02·47-s + 2/7·49-s + 0.274·53-s − 0.269·55-s − 1.82·59-s + 0.896·61-s − 0.248·65-s + 0.366·67-s + 0.237·71-s + 0.468·73-s − 0.683·77-s + 0.675·79-s + 0.987·83-s − 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.489664007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.489664007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 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
−7.84026014876681681021587958384, −7.46702465216147928235157860874, −6.69010947051879281826460974128, −5.73612487021766476100735713074, −5.03253543509403603999834853392, −4.73449929509122549076862611211, −3.55373904919237980185676158916, −2.61952242497891973289578598107, −1.88431442359074989086821714120, −0.835302566642866644868358542125,
0.835302566642866644868358542125, 1.88431442359074989086821714120, 2.61952242497891973289578598107, 3.55373904919237980185676158916, 4.73449929509122549076862611211, 5.03253543509403603999834853392, 5.73612487021766476100735713074, 6.69010947051879281826460974128, 7.46702465216147928235157860874, 7.84026014876681681021587958384
