| L(s) = 1 | + 2·5-s + 4·11-s + 4·13-s + 2·19-s + 4·23-s + 3·25-s − 4·31-s + 4·37-s + 8·43-s + 12·47-s − 12·49-s + 8·55-s + 8·59-s + 8·61-s + 8·65-s + 24·71-s − 12·73-s − 16·79-s + 4·83-s + 16·89-s + 4·95-s + 4·97-s − 4·101-s − 16·103-s + 16·107-s − 12·109-s + 8·113-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.20·11-s + 1.10·13-s + 0.458·19-s + 0.834·23-s + 3/5·25-s − 0.718·31-s + 0.657·37-s + 1.21·43-s + 1.75·47-s − 1.71·49-s + 1.07·55-s + 1.04·59-s + 1.02·61-s + 0.992·65-s + 2.84·71-s − 1.40·73-s − 1.80·79-s + 0.439·83-s + 1.69·89-s + 0.410·95-s + 0.406·97-s − 0.398·101-s − 1.57·103-s + 1.54·107-s − 1.14·109-s + 0.752·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46785600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46785600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.081464851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.081464851\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 52 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 190 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 16 T + 224 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 100 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.069147632572759034141889201044, −7.87075414684223591097796688509, −7.37370908730404227956166364003, −6.99468931375400305132437624302, −6.66675826127585502181994281964, −6.48994960882343430136223880575, −5.89196867383542511109516819655, −5.85478663909079894067307395207, −5.34200856921729974084035887292, −5.07090093319266017194171338144, −4.54470412197915323416869552356, −4.06561468946661811391035956125, −3.80771407560980882832889399690, −3.48737900461078471150091161812, −2.76315808331837961158412007014, −2.69055940658865735843236225723, −1.80004377745150944804416856342, −1.72254258018329980200785082085, −0.893125164363242432373359258361, −0.789941846978322263415504638994,
0.789941846978322263415504638994, 0.893125164363242432373359258361, 1.72254258018329980200785082085, 1.80004377745150944804416856342, 2.69055940658865735843236225723, 2.76315808331837961158412007014, 3.48737900461078471150091161812, 3.80771407560980882832889399690, 4.06561468946661811391035956125, 4.54470412197915323416869552356, 5.07090093319266017194171338144, 5.34200856921729974084035887292, 5.85478663909079894067307395207, 5.89196867383542511109516819655, 6.48994960882343430136223880575, 6.66675826127585502181994281964, 6.99468931375400305132437624302, 7.37370908730404227956166364003, 7.87075414684223591097796688509, 8.069147632572759034141889201044
