L(s) = 1 | + 5-s − 7-s + 4·11-s + 6·13-s + 6·17-s − 6·19-s − 23-s + 25-s + 2·29-s + 8·31-s − 35-s − 8·37-s + 10·41-s + 4·43-s − 12·47-s + 49-s + 6·53-s + 4·55-s + 10·59-s + 6·61-s + 6·65-s + 12·67-s − 2·71-s + 6·73-s − 4·77-s + 2·79-s + 4·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.20·11-s + 1.66·13-s + 1.45·17-s − 1.37·19-s − 0.208·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.169·35-s − 1.31·37-s + 1.56·41-s + 0.609·43-s − 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.539·55-s + 1.30·59-s + 0.768·61-s + 0.744·65-s + 1.46·67-s − 0.237·71-s + 0.702·73-s − 0.455·77-s + 0.225·79-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.338023718\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.338023718\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.59987393310650, −13.13981526772653, −12.65214539278725, −12.18564665405070, −11.68106932294607, −11.18202813117249, −10.66142376101919, −10.10243587354054, −9.809774931664962, −9.139059345831409, −8.633343127073512, −8.346472212515611, −7.740152611650026, −6.906592308591180, −6.458757216977368, −6.191633440357163, −5.672478529571531, −4.986510960692274, −4.270245953788664, −3.651948062980905, −3.479456133453510, −2.528525433227388, −1.905733958578325, −1.109775334910532, −0.7576964701976165,
0.7576964701976165, 1.109775334910532, 1.905733958578325, 2.528525433227388, 3.479456133453510, 3.651948062980905, 4.270245953788664, 4.986510960692274, 5.672478529571531, 6.191633440357163, 6.458757216977368, 6.906592308591180, 7.740152611650026, 8.346472212515611, 8.633343127073512, 9.139059345831409, 9.809774931664962, 10.10243587354054, 10.66142376101919, 11.18202813117249, 11.68106932294607, 12.18564665405070, 12.65214539278725, 13.13981526772653, 13.59987393310650