L(s) = 1 | + 2·2-s + 3·4-s − 2·7-s + 4·8-s + 9·11-s + 7·13-s − 4·14-s + 5·16-s + 2·17-s + 5·19-s + 18·22-s − 8·23-s − 10·25-s + 14·26-s − 6·28-s + 7·29-s + 6·31-s + 6·32-s + 4·34-s + 2·37-s + 10·38-s − 8·41-s − 3·43-s + 27·44-s − 16·46-s + 5·47-s + 2·49-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.755·7-s + 1.41·8-s + 2.71·11-s + 1.94·13-s − 1.06·14-s + 5/4·16-s + 0.485·17-s + 1.14·19-s + 3.83·22-s − 1.66·23-s − 2·25-s + 2.74·26-s − 1.13·28-s + 1.29·29-s + 1.07·31-s + 1.06·32-s + 0.685·34-s + 0.328·37-s + 1.62·38-s − 1.24·41-s − 0.457·43-s + 4.07·44-s − 2.35·46-s + 0.729·47-s + 2/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16112196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16112196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(11.00033849\) |
\(L(\frac12)\) |
\(\approx\) |
\(11.00033849\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 223 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 9 T + 39 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 7 T + 35 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 5 T + 41 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 67 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 62 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 59 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 97 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 79 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T + 117 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 139 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 122 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 129 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + T + 77 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 20 T + 226 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 206 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.452512580828103690275563107163, −8.327948808020895528790761162709, −7.75829950159078180381952489958, −7.52128082438125781523221355795, −6.78108393961842253342077941767, −6.65292165646271728631785750404, −6.29405210868210541701229249654, −6.13742942172902619036742078109, −5.66554143417133303229295296261, −5.54958076003078191977393141470, −4.54032896007714357708221393900, −4.49819863725997226427627388022, −3.85513249980711232195938008799, −3.78655078297298587025780136077, −3.30996906844965300312688807378, −3.16256737062436172329747616053, −2.24763094074381032511929667268, −1.74274490168439221789530784475, −1.29875341973171179208727592951, −0.813117710041396841667166824850,
0.813117710041396841667166824850, 1.29875341973171179208727592951, 1.74274490168439221789530784475, 2.24763094074381032511929667268, 3.16256737062436172329747616053, 3.30996906844965300312688807378, 3.78655078297298587025780136077, 3.85513249980711232195938008799, 4.49819863725997226427627388022, 4.54032896007714357708221393900, 5.54958076003078191977393141470, 5.66554143417133303229295296261, 6.13742942172902619036742078109, 6.29405210868210541701229249654, 6.65292165646271728631785750404, 6.78108393961842253342077941767, 7.52128082438125781523221355795, 7.75829950159078180381952489958, 8.327948808020895528790761162709, 8.452512580828103690275563107163