| L(s) = 1 | + 48·4-s + 868·11-s + 1.28e3·16-s + 1.46e3·19-s − 1.51e4·29-s + 4.35e3·31-s + 2.40e4·41-s + 4.16e4·44-s − 1.70e4·49-s − 1.03e4·59-s − 7.74e4·61-s + 1.22e4·64-s − 1.28e5·71-s + 7.01e4·76-s + 2.10e5·79-s − 1.30e5·89-s + 1.89e5·101-s + 3.05e5·109-s − 7.27e5·116-s + 2.42e5·121-s + 2.08e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 2.16·11-s + 5/4·16-s + 0.929·19-s − 3.34·29-s + 0.812·31-s + 2.23·41-s + 3.24·44-s − 1.01·49-s − 0.387·59-s − 2.66·61-s + 3/8·64-s − 3.03·71-s + 1.39·76-s + 3.78·79-s − 1.74·89-s + 1.84·101-s + 2.46·109-s − 5.02·116-s + 1.50·121-s + 1.21·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.446768566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.446768566\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p^{4} T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 17011 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 434 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 366817 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2068830 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 731 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4750186 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7582 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2175 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 52011814 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12040 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 292760245 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 434004870 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 803443386 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 5174 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 38717 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 1694916365 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 64472 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3754499086 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 105000 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7867072162 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 65376 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 980785 p^{2} T^{2} + p^{10} T^{4} \) |
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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
−11.77897734978769885806608190057, −11.15784565732796546426422114116, −10.92048885101646033875403853563, −10.26783365073773239997358307839, −9.401574193717058974833673186510, −9.355562418258033390930461458521, −8.903495075609622365856676722944, −7.71149413823051440840591205929, −7.68668666623897820020904613437, −7.09268761161577495451363039028, −6.53865552799869839414018766886, −5.91835408564323845345350733703, −5.85899817955017855650964339011, −4.71253525349099420711618876468, −4.03531330484900295970567952058, −3.42721139637693058686624114420, −2.86171506797837818787058134677, −1.76791616078745120525825255003, −1.63860312880416090180511276353, −0.67538304027871857389057474475,
0.67538304027871857389057474475, 1.63860312880416090180511276353, 1.76791616078745120525825255003, 2.86171506797837818787058134677, 3.42721139637693058686624114420, 4.03531330484900295970567952058, 4.71253525349099420711618876468, 5.85899817955017855650964339011, 5.91835408564323845345350733703, 6.53865552799869839414018766886, 7.09268761161577495451363039028, 7.68668666623897820020904613437, 7.71149413823051440840591205929, 8.903495075609622365856676722944, 9.355562418258033390930461458521, 9.401574193717058974833673186510, 10.26783365073773239997358307839, 10.92048885101646033875403853563, 11.15784565732796546426422114116, 11.77897734978769885806608190057
