| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5198400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5198400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.211039035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.211039035\) |
| \(L(\frac{3}{2})\) |
|
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
−9.064853322519666626477245508193, −8.968865208259580541075062751889, −8.402341790342959176459082174420, −7.974927001498641421293566208412, −7.62789721629505910991391924876, −7.12537956973785423526028238406, −6.81082477007118295873300825625, −6.71777962210647024934771792138, −6.08858926564127662291794786963, −5.68343894102053779376246710414, −4.99348905177960789715348605884, −4.94103167931776267396570125045, −4.53037028754578721107836215668, −4.12897135163360624663474684641, −3.40523892506397326240237129128, −3.18082128905359673467485530016, −2.15066712598884637300419173657, −2.13197779677824330423367568462, −0.75969683485857750146538339690, −0.62731283227951110386030647188,
0.62731283227951110386030647188, 0.75969683485857750146538339690, 2.13197779677824330423367568462, 2.15066712598884637300419173657, 3.18082128905359673467485530016, 3.40523892506397326240237129128, 4.12897135163360624663474684641, 4.53037028754578721107836215668, 4.94103167931776267396570125045, 4.99348905177960789715348605884, 5.68343894102053779376246710414, 6.08858926564127662291794786963, 6.71777962210647024934771792138, 6.81082477007118295873300825625, 7.12537956973785423526028238406, 7.62789721629505910991391924876, 7.974927001498641421293566208412, 8.402341790342959176459082174420, 8.968865208259580541075062751889, 9.064853322519666626477245508193
