| L(s) = 1 | + 4-s − 3·5-s + 9-s − 3·11-s + 16-s − 6·19-s − 3·20-s + 4·25-s − 3·29-s + 10·31-s + 36-s + 12·41-s − 3·44-s − 3·45-s − 4·49-s + 9·55-s − 6·59-s + 16·61-s + 64-s − 9·71-s − 6·76-s − 2·79-s − 3·80-s − 8·81-s + 9·89-s + 18·95-s − 3·99-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.34·5-s + 1/3·9-s − 0.904·11-s + 1/4·16-s − 1.37·19-s − 0.670·20-s + 4/5·25-s − 0.557·29-s + 1.79·31-s + 1/6·36-s + 1.87·41-s − 0.452·44-s − 0.447·45-s − 4/7·49-s + 1.21·55-s − 0.781·59-s + 2.04·61-s + 1/8·64-s − 1.06·71-s − 0.688·76-s − 0.225·79-s − 0.335·80-s − 8/9·81-s + 0.953·89-s + 1.84·95-s − 0.301·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5856958784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5856958784\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 7 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 89 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
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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
−13.12629201823161730215632551035, −12.77978038682690811603907751033, −12.12462061109148388225220357108, −11.51959044376157204820634393232, −10.96581222542300546543217413733, −10.43360058383073541051440337968, −9.701139036768523738942555924982, −8.643755561385292296510754105134, −8.068899168563034845611018261769, −7.54335577564449427577768867852, −6.77565335418431214450302625944, −5.91364375610394068414214605262, −4.70053508669332129152077478238, −3.95180666919658488912421076947, −2.65340032178954426050786831781,
2.65340032178954426050786831781, 3.95180666919658488912421076947, 4.70053508669332129152077478238, 5.91364375610394068414214605262, 6.77565335418431214450302625944, 7.54335577564449427577768867852, 8.068899168563034845611018261769, 8.643755561385292296510754105134, 9.701139036768523738942555924982, 10.43360058383073541051440337968, 10.96581222542300546543217413733, 11.51959044376157204820634393232, 12.12462061109148388225220357108, 12.77978038682690811603907751033, 13.12629201823161730215632551035
