| L(s) = 1 | − 32·4-s + 31·5-s − 162·9-s + 466·11-s + 768·16-s + 722·19-s − 992·20-s + 336·25-s + 5.18e3·36-s − 1.49e4·44-s − 5.02e3·45-s + 527·49-s + 1.44e4·55-s − 6.33e3·61-s − 1.63e4·64-s − 2.31e4·76-s + 2.38e4·80-s + 1.96e4·81-s + 2.23e4·95-s − 7.54e4·99-s − 1.07e4·100-s + 1.99e4·101-s + 1.33e5·121-s − 8.95e3·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.23·5-s − 2·9-s + 3.85·11-s + 3·16-s + 2·19-s − 2.47·20-s + 0.537·25-s + 4·36-s − 7.70·44-s − 2.47·45-s + 0.219·49-s + 4.77·55-s − 1.70·61-s − 4·64-s − 4·76-s + 3.71·80-s + 3·81-s + 2.47·95-s − 7.70·99-s − 1.07·100-s + 1.96·101-s + 9.12·121-s − 0.573·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.009967728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.009967728\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 - 31 T + p^{4} T^{2} \) |
| 19 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 73 T + p^{4} T^{2} )( 1 + 73 T + p^{4} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 233 T + p^{4} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 353 T + p^{4} T^{2} )( 1 + 353 T + p^{4} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 158 T + p^{4} T^{2} )( 1 + 158 T + p^{4} T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 3527 T + p^{4} T^{2} )( 1 + 3527 T + p^{4} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 1207 T + p^{4} T^{2} )( 1 + 1207 T + p^{4} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 3167 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10033 T + p^{4} T^{2} )( 1 + 10033 T + p^{4} T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 5678 T + p^{4} T^{2} )( 1 + 5678 T + p^{4} T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
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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
−13.56366994200087445734826103036, −13.48559050732049559766878704582, −12.31052597209563439822921729227, −11.98278851078587289704862415083, −11.63790066229888743099447853155, −10.87880375862105561651700749361, −9.724669212263119667036604910974, −9.649954713760270084685920186065, −9.047054067518865911720189175949, −8.947560471711389729707366350072, −8.350922455075624826772674772696, −7.33206726668424913589399642107, −6.21120762413494093309307430008, −6.02915077023168424855130583504, −5.34881228310999129018216680068, −4.58558262019155727278335145272, −3.68945884836137828924786831286, −3.24553704052601245298815987119, −1.47765122615129073750647223808, −0.78481408612752177089805692941,
0.78481408612752177089805692941, 1.47765122615129073750647223808, 3.24553704052601245298815987119, 3.68945884836137828924786831286, 4.58558262019155727278335145272, 5.34881228310999129018216680068, 6.02915077023168424855130583504, 6.21120762413494093309307430008, 7.33206726668424913589399642107, 8.350922455075624826772674772696, 8.947560471711389729707366350072, 9.047054067518865911720189175949, 9.649954713760270084685920186065, 9.724669212263119667036604910974, 10.87880375862105561651700749361, 11.63790066229888743099447853155, 11.98278851078587289704862415083, 12.31052597209563439822921729227, 13.48559050732049559766878704582, 13.56366994200087445734826103036
