| L(s) = 1 | + 4·7-s + 5·9-s − 4·17-s + 2·23-s + 9·25-s + 14·31-s + 16·41-s + 16·47-s − 2·49-s + 20·63-s + 6·71-s − 8·73-s − 20·79-s + 16·81-s − 30·89-s − 14·97-s + 32·103-s + 18·113-s − 16·119-s − 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 20·153-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 5/3·9-s − 0.970·17-s + 0.417·23-s + 9/5·25-s + 2.51·31-s + 2.49·41-s + 2.33·47-s − 2/7·49-s + 2.51·63-s + 0.712·71-s − 0.936·73-s − 2.25·79-s + 16/9·81-s − 3.17·89-s − 1.42·97-s + 3.15·103-s + 1.69·113-s − 1.46·119-s − 0.0909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.61·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7929856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7929856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.034038306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.034038306\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−8.858569439647824559752818618109, −8.687592205667708506770154565991, −8.162704554022076151146745808998, −7.916878178629849818679257198326, −7.37157487360820519231106280778, −7.20266238781882351223882659480, −6.72927831674085923018263854945, −6.53660396588792702119274524157, −5.69417433013758956753560108064, −5.69133036420126825100930033639, −4.86319187009344073465388727768, −4.45298646865807577967014499988, −4.38891509824061165744603971286, −4.28839229713789975173015221894, −3.26774570394034284914638144952, −2.79179970004899877947433686724, −2.32943227969259500862063844775, −1.77049251638321865536227064279, −1.03770031424667030589951552802, −0.951595236594138342146761919373,
0.951595236594138342146761919373, 1.03770031424667030589951552802, 1.77049251638321865536227064279, 2.32943227969259500862063844775, 2.79179970004899877947433686724, 3.26774570394034284914638144952, 4.28839229713789975173015221894, 4.38891509824061165744603971286, 4.45298646865807577967014499988, 4.86319187009344073465388727768, 5.69133036420126825100930033639, 5.69417433013758956753560108064, 6.53660396588792702119274524157, 6.72927831674085923018263854945, 7.20266238781882351223882659480, 7.37157487360820519231106280778, 7.916878178629849818679257198326, 8.162704554022076151146745808998, 8.687592205667708506770154565991, 8.858569439647824559752818618109
