| L(s) = 1 | + 4·4-s − 4·7-s − 2·13-s + 12·16-s + 15·19-s − 5·25-s − 16·28-s + 15·31-s − 8·43-s + 9·49-s − 8·52-s − 61-s + 32·64-s − 21·67-s + 24·73-s + 60·76-s − 13·79-s + 8·91-s − 9·97-s − 20·100-s − 20·103-s + 36·109-s − 48·112-s − 11·121-s + 60·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2·4-s − 1.51·7-s − 0.554·13-s + 3·16-s + 3.44·19-s − 25-s − 3.02·28-s + 2.69·31-s − 1.21·43-s + 9/7·49-s − 1.10·52-s − 0.128·61-s + 4·64-s − 2.56·67-s + 2.80·73-s + 6.88·76-s − 1.46·79-s + 0.838·91-s − 0.913·97-s − 2·100-s − 1.97·103-s + 3.44·109-s − 4.53·112-s − 121-s + 5.38·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.191980188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.191980188\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.21542044498469552153961220043, −10.03090772372242502431819257984, −9.661705528373869945269933876820, −9.635023765519565968415240702370, −8.723162376681562215409651022055, −8.142200196690630770983997233034, −7.59583823397693846133706873422, −7.50071049439439988675137716864, −6.92177530320279608055596495996, −6.67140302187334630284381890088, −6.03362407839614074902443083005, −5.91367053766579945903153450923, −5.27324062478636023442147678404, −4.76036566512188688452287553418, −3.69957392534034410954236123761, −3.34539734953464998389171891832, −2.80543961373722632397572010385, −2.66819301919653587642606797314, −1.61389568997971843237903016251, −0.917135504782075917951374909555,
0.917135504782075917951374909555, 1.61389568997971843237903016251, 2.66819301919653587642606797314, 2.80543961373722632397572010385, 3.34539734953464998389171891832, 3.69957392534034410954236123761, 4.76036566512188688452287553418, 5.27324062478636023442147678404, 5.91367053766579945903153450923, 6.03362407839614074902443083005, 6.67140302187334630284381890088, 6.92177530320279608055596495996, 7.50071049439439988675137716864, 7.59583823397693846133706873422, 8.142200196690630770983997233034, 8.723162376681562215409651022055, 9.635023765519565968415240702370, 9.661705528373869945269933876820, 10.03090772372242502431819257984, 10.21542044498469552153961220043
