| L(s) = 1 | − 2·3-s + 4-s − 2·5-s − 2·7-s + 3·9-s − 2·12-s + 4·15-s − 3·16-s + 10·17-s − 10·19-s − 2·20-s + 4·21-s + 2·23-s − 2·25-s − 4·27-s − 2·28-s − 4·31-s + 4·35-s + 3·36-s + 4·41-s − 2·43-s − 6·45-s − 8·47-s + 6·48-s − 6·49-s − 20·51-s − 6·53-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s − 0.894·5-s − 0.755·7-s + 9-s − 0.577·12-s + 1.03·15-s − 3/4·16-s + 2.42·17-s − 2.29·19-s − 0.447·20-s + 0.872·21-s + 0.417·23-s − 2/5·25-s − 0.769·27-s − 0.377·28-s − 0.718·31-s + 0.676·35-s + 1/2·36-s + 0.624·41-s − 0.304·43-s − 0.894·45-s − 1.16·47-s + 0.866·48-s − 6/7·49-s − 2.80·51-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69705801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69705801 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 54 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 31 | $C_4$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 110 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 138 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 174 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 8 T + 190 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.63436152986578918415718280521, −7.04190023959609658635490568859, −7.03065690298663291311830354517, −6.54387192865241842667649749611, −6.32120398852489942536174298299, −5.86496611907683710163698033170, −5.72522250813225253838965857790, −5.24145804912378940565790313594, −4.79615829444638120225415238612, −4.52893266977304399700532300353, −3.97564231268877327011775857777, −3.83380231487870749457107606245, −3.34573086731022774490455207722, −2.93559839418215937160557230089, −2.50576568889202376938850684215, −1.79816907198708919813626148859, −1.54217691887302620801748395961, −0.836760849777885838778444324653, 0, 0,
0.836760849777885838778444324653, 1.54217691887302620801748395961, 1.79816907198708919813626148859, 2.50576568889202376938850684215, 2.93559839418215937160557230089, 3.34573086731022774490455207722, 3.83380231487870749457107606245, 3.97564231268877327011775857777, 4.52893266977304399700532300353, 4.79615829444638120225415238612, 5.24145804912378940565790313594, 5.72522250813225253838965857790, 5.86496611907683710163698033170, 6.32120398852489942536174298299, 6.54387192865241842667649749611, 7.03065690298663291311830354517, 7.04190023959609658635490568859, 7.63436152986578918415718280521
