L(s) = 1 | + 2·3-s − 2·5-s + 3·9-s − 4·13-s − 4·15-s − 4·17-s + 2·19-s − 8·23-s + 3·25-s + 4·27-s + 4·29-s − 4·37-s − 8·39-s + 4·41-s − 6·45-s − 8·47-s − 6·49-s − 8·51-s − 4·53-s + 4·57-s − 16·59-s − 4·61-s + 8·65-s − 8·67-s − 16·69-s − 16·71-s − 12·73-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.894·5-s + 9-s − 1.10·13-s − 1.03·15-s − 0.970·17-s + 0.458·19-s − 1.66·23-s + 3/5·25-s + 0.769·27-s + 0.742·29-s − 0.657·37-s − 1.28·39-s + 0.624·41-s − 0.894·45-s − 1.16·47-s − 6/7·49-s − 1.12·51-s − 0.549·53-s + 0.529·57-s − 2.08·59-s − 0.512·61-s + 0.992·65-s − 0.977·67-s − 1.92·69-s − 1.89·71-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 118 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 270 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 190 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.148116820842654461544170173428, −7.71771676946841932791460204748, −7.49314161444815504826321043461, −7.37140019889219946640748657571, −6.70248901848599863448522470964, −6.52827737111957505618378066263, −5.95867221759944079146496900026, −5.68227512687301740409495495150, −4.80139748531886841141519002397, −4.77533704110030025231436095610, −4.31523442756030364869962938077, −4.11538801448683417180965887870, −3.33388712717912145730446938112, −3.23250443894761546937962744477, −2.69757400236581797602947353197, −2.35173095112769002178796015819, −1.61727936024371041873255022653, −1.40301965935946598728794464790, 0, 0,
1.40301965935946598728794464790, 1.61727936024371041873255022653, 2.35173095112769002178796015819, 2.69757400236581797602947353197, 3.23250443894761546937962744477, 3.33388712717912145730446938112, 4.11538801448683417180965887870, 4.31523442756030364869962938077, 4.77533704110030025231436095610, 4.80139748531886841141519002397, 5.68227512687301740409495495150, 5.95867221759944079146496900026, 6.52827737111957505618378066263, 6.70248901848599863448522470964, 7.37140019889219946640748657571, 7.49314161444815504826321043461, 7.71771676946841932791460204748, 8.148116820842654461544170173428