L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s + 6·11-s + 4·14-s + 16-s + 4·17-s − 19-s + 6·22-s − 4·23-s + 4·28-s + 10·29-s − 2·31-s + 32-s + 4·34-s + 4·37-s − 38-s − 10·41-s + 12·43-s + 6·44-s − 4·46-s + 9·49-s − 6·53-s + 4·56-s + 10·58-s − 4·59-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 1.80·11-s + 1.06·14-s + 1/4·16-s + 0.970·17-s − 0.229·19-s + 1.27·22-s − 0.834·23-s + 0.755·28-s + 1.85·29-s − 0.359·31-s + 0.176·32-s + 0.685·34-s + 0.657·37-s − 0.162·38-s − 1.56·41-s + 1.82·43-s + 0.904·44-s − 0.589·46-s + 9/7·49-s − 0.824·53-s + 0.534·56-s + 1.31·58-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.113729443\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.113729443\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.82440941702371093882599632825, −6.97928784799660693830124452550, −6.31664073410883494640406913040, −5.69674485102230251419580433097, −4.83643126536051389174384294154, −4.32162586882184529844881012567, −3.70382371515602142496458478975, −2.70897349007062590009475662712, −1.64783639851140525740072906287, −1.15440653027577646177790527258,
1.15440653027577646177790527258, 1.64783639851140525740072906287, 2.70897349007062590009475662712, 3.70382371515602142496458478975, 4.32162586882184529844881012567, 4.83643126536051389174384294154, 5.69674485102230251419580433097, 6.31664073410883494640406913040, 6.97928784799660693830124452550, 7.82440941702371093882599632825