| L(s) = 1 | − 4-s + 4·7-s − 13-s + 19-s − 25-s − 4·28-s + 31-s + 37-s − 2·43-s + 10·49-s + 52-s − 2·61-s + 64-s − 2·67-s − 2·73-s − 76-s + 79-s − 4·91-s − 2·97-s + 100-s − 2·103-s − 2·109-s − 121-s − 124-s + 127-s + 131-s + 4·133-s + ⋯ |
| L(s) = 1 | − 4-s + 4·7-s − 13-s + 19-s − 25-s − 4·28-s + 31-s + 37-s − 2·43-s + 10·49-s + 52-s − 2·61-s + 64-s − 2·67-s − 2·73-s − 76-s + 79-s − 4·91-s − 2·97-s + 100-s − 2·103-s − 2·109-s − 121-s − 124-s + 127-s + 131-s + 4·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1108809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1108809 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.217440026\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.217440026\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
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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
−10.23704914992218745391626942060, −10.01095003093223815659584033864, −9.216019771550208950510469717174, −9.191498616977325427273532421756, −8.408233962665429344204001371610, −8.330477287750654184636961315417, −7.82756707894427780908949543687, −7.69017942775142309345399351084, −7.26632587461682163967696025736, −6.61958537202565282917423637121, −5.61790621671738045036693127095, −5.54855374854932060318400800243, −4.95301453467013961885352055369, −4.72247173131924966925870251894, −4.31787549763402384801120994026, −4.12096246784870526425075418661, −2.97531727252643223703800178133, −2.33825696699232122117955207074, −1.59512119726909521771722121501, −1.32258396653532471848543228392,
1.32258396653532471848543228392, 1.59512119726909521771722121501, 2.33825696699232122117955207074, 2.97531727252643223703800178133, 4.12096246784870526425075418661, 4.31787549763402384801120994026, 4.72247173131924966925870251894, 4.95301453467013961885352055369, 5.54855374854932060318400800243, 5.61790621671738045036693127095, 6.61958537202565282917423637121, 7.26632587461682163967696025736, 7.69017942775142309345399351084, 7.82756707894427780908949543687, 8.330477287750654184636961315417, 8.408233962665429344204001371610, 9.191498616977325427273532421756, 9.216019771550208950510469717174, 10.01095003093223815659584033864, 10.23704914992218745391626942060
