L(s) = 1 | − 2·3-s + 5-s − 2·7-s + 9-s − 13-s − 2·15-s + 5·17-s + 6·19-s + 4·21-s − 2·23-s − 4·25-s + 4·27-s − 9·29-s + 2·31-s − 2·35-s − 3·37-s + 2·39-s + 5·41-s + 45-s − 2·47-s − 3·49-s − 10·51-s + 9·53-s − 12·57-s − 8·59-s − 6·61-s − 2·63-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.277·13-s − 0.516·15-s + 1.21·17-s + 1.37·19-s + 0.872·21-s − 0.417·23-s − 4/5·25-s + 0.769·27-s − 1.67·29-s + 0.359·31-s − 0.338·35-s − 0.493·37-s + 0.320·39-s + 0.780·41-s + 0.149·45-s − 0.291·47-s − 3/7·49-s − 1.40·51-s + 1.23·53-s − 1.58·57-s − 1.04·59-s − 0.768·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.043042867788870716539415194167, −7.78091021638599658560987714578, −7.18510639953264999652502212264, −6.10386760784300470004944448119, −5.73468060896468798044264674509, −5.04523846649891070859056262216, −3.79587121545739932046191560595, −2.85249788770459737210058313708, −1.38254023963820058194153479990, 0,
1.38254023963820058194153479990, 2.85249788770459737210058313708, 3.79587121545739932046191560595, 5.04523846649891070859056262216, 5.73468060896468798044264674509, 6.10386760784300470004944448119, 7.18510639953264999652502212264, 7.78091021638599658560987714578, 9.043042867788870716539415194167