| L(s) = 1 | − 4·5-s + 5·9-s + 2·11-s − 8·19-s + 11·25-s − 2·29-s − 12·31-s − 20·41-s − 20·45-s − 49-s − 8·55-s + 12·59-s + 8·61-s − 32·71-s + 22·79-s + 16·81-s − 24·89-s + 32·95-s + 10·99-s − 30·109-s − 19·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 5/3·9-s + 0.603·11-s − 1.83·19-s + 11/5·25-s − 0.371·29-s − 2.15·31-s − 3.12·41-s − 2.98·45-s − 1/7·49-s − 1.07·55-s + 1.56·59-s + 1.02·61-s − 3.79·71-s + 2.47·79-s + 16/9·81-s − 2.54·89-s + 3.28·95-s + 1.00·99-s − 2.87·109-s − 1.72·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5017600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5695330004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5695330004\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 167 T^{2} + p^{2} T^{4} \) |
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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
−9.093539465799854060103428884620, −8.817657413218112451353241511906, −8.417383794736578545521292719953, −8.058417322185325920710096687510, −7.68793383710606547363503198831, −7.24982985865202971765063356171, −6.93655542602082511837804887232, −6.64417896302252110208382440785, −6.45597334346965511796262917175, −5.39078557897195606851427934222, −5.35621408071497572020905633756, −4.62272669291462483041148488468, −4.22736966864715661688539858549, −3.98573084281879101764207324267, −3.69132277762757181282166305959, −3.22355143435925611147062859124, −2.43008637385855123559522439640, −1.67886796995363066051158480688, −1.41627846007304574185549093111, −0.26782730409805383087605093350,
0.26782730409805383087605093350, 1.41627846007304574185549093111, 1.67886796995363066051158480688, 2.43008637385855123559522439640, 3.22355143435925611147062859124, 3.69132277762757181282166305959, 3.98573084281879101764207324267, 4.22736966864715661688539858549, 4.62272669291462483041148488468, 5.35621408071497572020905633756, 5.39078557897195606851427934222, 6.45597334346965511796262917175, 6.64417896302252110208382440785, 6.93655542602082511837804887232, 7.24982985865202971765063356171, 7.68793383710606547363503198831, 8.058417322185325920710096687510, 8.417383794736578545521292719953, 8.817657413218112451353241511906, 9.093539465799854060103428884620
