| L(s) = 1 | + 4-s − 5-s − 6·9-s + 8·11-s + 16-s − 20-s + 25-s + 12·29-s + 16·31-s − 6·36-s + 4·41-s + 8·44-s + 6·45-s + 49-s − 8·55-s − 16·59-s − 28·61-s + 64-s − 32·71-s − 16·79-s − 80-s + 27·81-s + 20·89-s − 48·99-s + 100-s − 12·101-s + 12·109-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.447·5-s − 2·9-s + 2.41·11-s + 1/4·16-s − 0.223·20-s + 1/5·25-s + 2.22·29-s + 2.87·31-s − 36-s + 0.624·41-s + 1.20·44-s + 0.894·45-s + 1/7·49-s − 1.07·55-s − 2.08·59-s − 3.58·61-s + 1/8·64-s − 3.79·71-s − 1.80·79-s − 0.111·80-s + 3·81-s + 2.11·89-s − 4.82·99-s + 1/10·100-s − 1.19·101-s + 1.14·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.246042932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.246042932\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( 1 + T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
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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
−10.68041607351861279913180531310, −10.37599431086570880625729241811, −9.460094722609108544443052630434, −8.977646092372605556420891081568, −8.629572853345107589851782221455, −8.089729535357517933440155073370, −7.45420248729985818641828193491, −6.56718920518778472538358027386, −6.17616144092315208416665412995, −6.01430128602422585899125610915, −4.63923341949934829303628265619, −4.36823223624486251008730105784, −3.00381372509091040158022668795, −2.98711940319117381089884417433, −1.27008084899723889248060652436,
1.27008084899723889248060652436, 2.98711940319117381089884417433, 3.00381372509091040158022668795, 4.36823223624486251008730105784, 4.63923341949934829303628265619, 6.01430128602422585899125610915, 6.17616144092315208416665412995, 6.56718920518778472538358027386, 7.45420248729985818641828193491, 8.089729535357517933440155073370, 8.629572853345107589851782221455, 8.977646092372605556420891081568, 9.460094722609108544443052630434, 10.37599431086570880625729241811, 10.68041607351861279913180531310
