L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s + 4·13-s − 15-s + 4·17-s + 8·19-s − 2·21-s − 4·23-s + 25-s + 27-s − 2·29-s − 8·31-s + 2·35-s − 2·37-s + 4·39-s + 2·41-s − 6·43-s − 45-s − 12·47-s − 3·49-s + 4·51-s + 6·53-s + 8·57-s + 12·59-s − 10·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.10·13-s − 0.258·15-s + 0.970·17-s + 1.83·19-s − 0.436·21-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.338·35-s − 0.328·37-s + 0.640·39-s + 0.312·41-s − 0.914·43-s − 0.149·45-s − 1.75·47-s − 3/7·49-s + 0.560·51-s + 0.824·53-s + 1.05·57-s + 1.56·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.52075896214349, −14.72555274464321, −14.57045908197571, −13.69511608471118, −13.43138828088508, −12.91663096207999, −12.18073247695431, −11.79702238558277, −11.23797361656267, −10.47864917837314, −10.01638822388093, −9.352653064971938, −9.100149552189961, −8.124728869639667, −7.948918045167400, −7.184012897884133, −6.727709835969544, −5.837523239764515, −5.489305525117142, −4.611625292462804, −3.621370860083790, −3.515352155781278, −2.918746780119483, −1.780142730161228, −1.138961261431155, 0,
1.138961261431155, 1.780142730161228, 2.918746780119483, 3.515352155781278, 3.621370860083790, 4.611625292462804, 5.489305525117142, 5.837523239764515, 6.727709835969544, 7.184012897884133, 7.948918045167400, 8.124728869639667, 9.100149552189961, 9.352653064971938, 10.01638822388093, 10.47864917837314, 11.23797361656267, 11.79702238558277, 12.18073247695431, 12.91663096207999, 13.43138828088508, 13.69511608471118, 14.57045908197571, 14.72555274464321, 15.52075896214349