| L(s) = 1 | − 5-s − 2·7-s − 3·11-s − 5·13-s + 3·17-s − 4·19-s − 9·23-s + 25-s − 3·29-s − 5·31-s + 2·35-s + 10·37-s − 43-s + 9·47-s − 3·49-s − 12·53-s + 3·55-s − 12·59-s − 2·61-s + 5·65-s − 4·67-s + 12·71-s − 10·73-s + 6·77-s + 13·79-s − 6·83-s − 3·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.904·11-s − 1.38·13-s + 0.727·17-s − 0.917·19-s − 1.87·23-s + 1/5·25-s − 0.557·29-s − 0.898·31-s + 0.338·35-s + 1.64·37-s − 0.152·43-s + 1.31·47-s − 3/7·49-s − 1.64·53-s + 0.404·55-s − 1.56·59-s − 0.256·61-s + 0.620·65-s − 0.488·67-s + 1.42·71-s − 1.17·73-s + 0.683·77-s + 1.46·79-s − 0.658·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4751940710\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4751940710\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−7.78606734654073813602083601909, −7.27252964105726081799610550986, −6.29213279651374617112655497765, −5.82313513812231665875351105896, −4.92564264371569373065645889541, −4.26347796467935794261111953791, −3.45078876610182730473378817032, −2.65782181453904725650079273202, −1.92567694146375915716031269078, −0.31252346122714268383195975984,
0.31252346122714268383195975984, 1.92567694146375915716031269078, 2.65782181453904725650079273202, 3.45078876610182730473378817032, 4.26347796467935794261111953791, 4.92564264371569373065645889541, 5.82313513812231665875351105896, 6.29213279651374617112655497765, 7.27252964105726081799610550986, 7.78606734654073813602083601909
