| L(s) = 1 | + 2-s + 2·3-s + 2·4-s − 2·5-s + 2·6-s + 4·7-s + 5·8-s + 3·9-s − 2·10-s − 2·11-s + 4·12-s + 4·14-s − 4·15-s + 5·16-s − 2·17-s + 3·18-s + 6·19-s − 4·20-s + 8·21-s − 2·22-s + 6·23-s + 10·24-s + 3·25-s + 10·27-s + 8·28-s − 2·29-s − 4·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 4-s − 0.894·5-s + 0.816·6-s + 1.51·7-s + 1.76·8-s + 9-s − 0.632·10-s − 0.603·11-s + 1.15·12-s + 1.06·14-s − 1.03·15-s + 5/4·16-s − 0.485·17-s + 0.707·18-s + 1.37·19-s − 0.894·20-s + 1.74·21-s − 0.426·22-s + 1.25·23-s + 2.04·24-s + 3/5·25-s + 1.92·27-s + 1.51·28-s − 0.371·29-s − 0.730·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.686564188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.686564188\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−10.37510494467091216201509006902, −10.26353700376450239156539179802, −9.497346536250884790438189541788, −8.960876002228204328301169517205, −8.504852953594261532333165453385, −8.374634293658928477650611957747, −7.51920171933055205115708108957, −7.49049183180776436878794037608, −7.09877311076717960323820402896, −7.07433239118255909087030325664, −5.70649782632833034290236070118, −5.60041402331262252614430599045, −4.73761507855249096565992207991, −4.63811229739031508824823749501, −4.16876264154762902278061351225, −3.36072748806479364920249046470, −3.12366061136478642687834164920, −2.34739667155265461468947618331, −1.73651162985269719514070159285, −1.23267521026612391642288385797,
1.23267521026612391642288385797, 1.73651162985269719514070159285, 2.34739667155265461468947618331, 3.12366061136478642687834164920, 3.36072748806479364920249046470, 4.16876264154762902278061351225, 4.63811229739031508824823749501, 4.73761507855249096565992207991, 5.60041402331262252614430599045, 5.70649782632833034290236070118, 7.07433239118255909087030325664, 7.09877311076717960323820402896, 7.49049183180776436878794037608, 7.51920171933055205115708108957, 8.374634293658928477650611957747, 8.504852953594261532333165453385, 8.960876002228204328301169517205, 9.497346536250884790438189541788, 10.26353700376450239156539179802, 10.37510494467091216201509006902
