L(s) = 1 | + 4·3-s − 2·5-s + 8·9-s − 4·11-s + 4·13-s − 8·15-s − 8·17-s + 4·19-s + 8·23-s + 3·25-s + 12·27-s − 4·29-s − 16·33-s − 4·37-s + 16·39-s − 8·41-s + 12·43-s − 16·45-s + 16·47-s − 32·51-s − 4·53-s + 8·55-s + 16·57-s + 20·59-s + 4·61-s − 8·65-s + 8·67-s + ⋯ |
L(s) = 1 | + 2.30·3-s − 0.894·5-s + 8/3·9-s − 1.20·11-s + 1.10·13-s − 2.06·15-s − 1.94·17-s + 0.917·19-s + 1.66·23-s + 3/5·25-s + 2.30·27-s − 0.742·29-s − 2.78·33-s − 0.657·37-s + 2.56·39-s − 1.24·41-s + 1.82·43-s − 2.38·45-s + 2.33·47-s − 4.48·51-s − 0.549·53-s + 1.07·55-s + 2.11·57-s + 2.60·59-s + 0.512·61-s − 0.992·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.380432687\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.380432687\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 114 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 20 T + 216 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 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 | $D_{4}$ | \( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 208 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 166 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 152 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 320 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−8.510681357508977944207093342558, −8.498721180093561669510885365990, −7.907096097715219973485383741831, −7.78553553643322768018561132562, −7.20501363643117499511727013126, −7.15178722084264799934689833095, −6.60315553458388563385291702998, −6.31317563288933886098026474916, −5.39500350223783840427539990003, −5.24575071850530207553389623313, −4.90164310537265708096105500017, −4.07831984226271294562981588852, −3.91701709551788122537831939669, −3.72188982284049364514294833987, −3.01900199660792516533741699049, −2.85082410284428121753506914211, −2.30387834697612667523824269756, −2.13805393770224068609389708092, −1.19358818603872925223689358355, −0.60511987838352224882598091851,
0.60511987838352224882598091851, 1.19358818603872925223689358355, 2.13805393770224068609389708092, 2.30387834697612667523824269756, 2.85082410284428121753506914211, 3.01900199660792516533741699049, 3.72188982284049364514294833987, 3.91701709551788122537831939669, 4.07831984226271294562981588852, 4.90164310537265708096105500017, 5.24575071850530207553389623313, 5.39500350223783840427539990003, 6.31317563288933886098026474916, 6.60315553458388563385291702998, 7.15178722084264799934689833095, 7.20501363643117499511727013126, 7.78553553643322768018561132562, 7.907096097715219973485383741831, 8.498721180093561669510885365990, 8.510681357508977944207093342558