L(s) = 1 | − 2·3-s + 4·4-s + 4·5-s − 3·9-s − 8·12-s − 8·15-s + 12·16-s + 6·17-s − 10·19-s + 16·20-s + 2·23-s + 2·25-s + 14·27-s + 2·29-s + 14·31-s − 12·36-s − 6·37-s + 20·41-s − 12·45-s − 24·48-s − 6·49-s − 12·51-s + 20·57-s − 32·60-s + 32·64-s − 26·67-s + 24·68-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 2·4-s + 1.78·5-s − 9-s − 2.30·12-s − 2.06·15-s + 3·16-s + 1.45·17-s − 2.29·19-s + 3.57·20-s + 0.417·23-s + 2/5·25-s + 2.69·27-s + 0.371·29-s + 2.51·31-s − 2·36-s − 0.986·37-s + 3.12·41-s − 1.78·45-s − 3.46·48-s − 6/7·49-s − 1.68·51-s + 2.64·57-s − 4.13·60-s + 4·64-s − 3.17·67-s + 2.91·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121801 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.239363501\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.239363501\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 349 | $C_2$ | \( 1 + 26 T + p T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 138 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
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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
−11.86343872111650149906984348679, −11.12367727251626748001177459835, −10.73447402918863351155831343460, −10.62362697135270406663297660139, −10.00322431469670045263320342590, −9.839257282582249937589771904985, −8.852948300127122231453789771735, −8.502230174053779631645065150992, −7.82550652207175712021595156171, −7.36130926723663591819888226407, −6.44393701970230526130647023645, −6.32078027152616484061343821959, −5.93068955907541386639922858133, −5.82785080793056503886606525470, −5.14619659011423115373513767235, −4.31393725584197670499754263938, −2.88551686569692652085333624868, −2.83570918859238160672004619638, −2.04582351986007842577548054606, −1.20313764724857323949867033725,
1.20313764724857323949867033725, 2.04582351986007842577548054606, 2.83570918859238160672004619638, 2.88551686569692652085333624868, 4.31393725584197670499754263938, 5.14619659011423115373513767235, 5.82785080793056503886606525470, 5.93068955907541386639922858133, 6.32078027152616484061343821959, 6.44393701970230526130647023645, 7.36130926723663591819888226407, 7.82550652207175712021595156171, 8.502230174053779631645065150992, 8.852948300127122231453789771735, 9.839257282582249937589771904985, 10.00322431469670045263320342590, 10.62362697135270406663297660139, 10.73447402918863351155831343460, 11.12367727251626748001177459835, 11.86343872111650149906984348679