L(s) = 1 | + 2·2-s + 3·4-s + 3·5-s + 4·8-s + 6·10-s − 6·11-s + 2·13-s + 5·16-s − 6·17-s − 7·19-s + 9·20-s − 12·22-s + 3·23-s + 5·25-s + 4·26-s + 6·29-s − 4·31-s + 6·32-s − 12·34-s − 2·37-s − 14·38-s + 12·40-s − 2·43-s − 18·44-s + 6·46-s + 10·50-s + 6·52-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.34·5-s + 1.41·8-s + 1.89·10-s − 1.80·11-s + 0.554·13-s + 5/4·16-s − 1.45·17-s − 1.60·19-s + 2.01·20-s − 2.55·22-s + 0.625·23-s + 25-s + 0.784·26-s + 1.11·29-s − 0.718·31-s + 1.06·32-s − 2.05·34-s − 0.328·37-s − 2.27·38-s + 1.89·40-s − 0.304·43-s − 2.71·44-s + 0.884·46-s + 1.41·50-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7001316 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.518876706\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.518876706\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.281555630135559416055333377262, −8.582802745365980176714549917598, −8.123721029730607122251098581030, −8.123063428493459659180928151609, −7.24043210112474355261780469862, −6.87844093293021943315136714684, −6.76896785249808213671159573099, −6.14115125480109365377394483201, −5.98342890635902755421339175316, −5.47657645947162229343466478161, −5.21566168260829610499324672835, −4.72150674143783035808953228403, −4.45053261262822048545312171887, −3.99476402641683646171195354192, −3.31419653659577975557109894294, −2.80735051271798484923763990792, −2.54395127821462212286025034497, −1.91200897076389879440871610987, −1.81132020339380924657786265295, −0.59350255717347315057394342410,
0.59350255717347315057394342410, 1.81132020339380924657786265295, 1.91200897076389879440871610987, 2.54395127821462212286025034497, 2.80735051271798484923763990792, 3.31419653659577975557109894294, 3.99476402641683646171195354192, 4.45053261262822048545312171887, 4.72150674143783035808953228403, 5.21566168260829610499324672835, 5.47657645947162229343466478161, 5.98342890635902755421339175316, 6.14115125480109365377394483201, 6.76896785249808213671159573099, 6.87844093293021943315136714684, 7.24043210112474355261780469862, 8.123063428493459659180928151609, 8.123721029730607122251098581030, 8.582802745365980176714549917598, 9.281555630135559416055333377262