| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 5·9-s − 10·13-s + 5·16-s + 8·17-s − 10·18-s + 6·19-s − 25-s + 20·26-s − 6·32-s − 16·34-s + 15·36-s − 12·38-s − 8·43-s + 14·47-s + 14·49-s + 2·50-s − 30·52-s − 2·53-s − 18·59-s + 7·64-s + 20·67-s + 24·68-s − 20·72-s + 18·76-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 5/3·9-s − 2.77·13-s + 5/4·16-s + 1.94·17-s − 2.35·18-s + 1.37·19-s − 1/5·25-s + 3.92·26-s − 1.06·32-s − 2.74·34-s + 5/2·36-s − 1.94·38-s − 1.21·43-s + 2.04·47-s + 2·49-s + 0.282·50-s − 4.16·52-s − 0.274·53-s − 2.34·59-s + 7/8·64-s + 2.44·67-s + 2.91·68-s − 2.35·72-s + 2.06·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7050842614\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7050842614\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 17 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 113 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 97 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 193 T^{2} + p^{2} T^{4} \) |
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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
−12.61603016955695792603578429928, −12.36265902060199657020855747812, −12.00723022924032505724382508335, −11.70416152727181662473690338497, −10.60233511927826921436502384283, −10.41184485445080752737826829294, −9.849399957671859471824036832083, −9.517224951693734740414588796622, −9.407551739869589008132607484608, −8.351184073823879274480511207488, −7.55489761825516621496059546104, −7.47260858679348041521214588364, −7.25067159768810349849370774076, −6.41214605013478270568276589895, −5.38363727405958443972670767649, −5.06925577053071545335203107003, −4.03515190005163281835807438076, −3.05268001774041829351668466912, −2.19185027504633142046958447704, −1.08657951342704377928838709233,
1.08657951342704377928838709233, 2.19185027504633142046958447704, 3.05268001774041829351668466912, 4.03515190005163281835807438076, 5.06925577053071545335203107003, 5.38363727405958443972670767649, 6.41214605013478270568276589895, 7.25067159768810349849370774076, 7.47260858679348041521214588364, 7.55489761825516621496059546104, 8.351184073823879274480511207488, 9.407551739869589008132607484608, 9.517224951693734740414588796622, 9.849399957671859471824036832083, 10.41184485445080752737826829294, 10.60233511927826921436502384283, 11.70416152727181662473690338497, 12.00723022924032505724382508335, 12.36265902060199657020855747812, 12.61603016955695792603578429928
