| L(s) = 1 | − 5-s + 2·7-s − 11-s + 13-s + 17-s + 4·19-s − 23-s + 25-s + 5·29-s + 31-s − 2·35-s + 6·37-s + 7·43-s − 7·47-s − 3·49-s + 12·53-s + 55-s + 4·59-s + 10·61-s − 65-s − 4·67-s − 12·71-s + 6·73-s − 2·77-s + 15·79-s − 2·83-s − 85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 0.301·11-s + 0.277·13-s + 0.242·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s + 0.928·29-s + 0.179·31-s − 0.338·35-s + 0.986·37-s + 1.06·43-s − 1.02·47-s − 3/7·49-s + 1.64·53-s + 0.134·55-s + 0.520·59-s + 1.28·61-s − 0.124·65-s − 0.488·67-s − 1.42·71-s + 0.702·73-s − 0.227·77-s + 1.68·79-s − 0.219·83-s − 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.655676292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.655676292\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.938016913173052801027619458454, −8.971544127814454878311880442588, −8.098215835619989410546011808466, −7.59623005890538420604075910258, −6.55466152525426843819318782499, −5.49385970565338845590944925790, −4.68426941255478221849145341689, −3.68561025556385408354635008380, −2.51888632026023556389235975025, −1.04466524673073804361911141068,
1.04466524673073804361911141068, 2.51888632026023556389235975025, 3.68561025556385408354635008380, 4.68426941255478221849145341689, 5.49385970565338845590944925790, 6.55466152525426843819318782499, 7.59623005890538420604075910258, 8.098215835619989410546011808466, 8.971544127814454878311880442588, 9.938016913173052801027619458454
