| L(s) = 1 | + 3·4-s − 2·5-s + 5·9-s + 5·16-s + 12·19-s − 6·20-s − 25-s − 14·29-s − 4·31-s + 15·36-s − 10·41-s − 10·45-s + 20·59-s − 14·61-s + 3·64-s − 4·71-s + 36·76-s + 4·79-s − 10·80-s + 16·81-s + 18·89-s − 24·95-s − 3·100-s − 18·101-s − 10·109-s − 42·116-s − 22·121-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.894·5-s + 5/3·9-s + 5/4·16-s + 2.75·19-s − 1.34·20-s − 1/5·25-s − 2.59·29-s − 0.718·31-s + 5/2·36-s − 1.56·41-s − 1.49·45-s + 2.60·59-s − 1.79·61-s + 3/8·64-s − 0.474·71-s + 4.12·76-s + 0.450·79-s − 1.11·80-s + 16/9·81-s + 1.90·89-s − 2.46·95-s − 0.299·100-s − 1.79·101-s − 0.957·109-s − 3.89·116-s − 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.987838820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.987838820\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−12.01571804133945693335686446554, −11.86674004305445302541064934117, −11.49802957873690594574763968636, −11.02037666658441009546729138770, −10.48053830642631756293273451889, −10.05806791301518125763005312962, −9.368509676824820831888843784883, −9.269489629824524181883533766445, −8.054107134123647440803206813455, −7.71201727717384931154224197875, −7.33008804031268611212863646820, −7.04964082746897053505966053312, −6.55840897780026521438484164195, −5.51763624678873988892869467209, −5.35919431520621974263615254112, −4.29928078103888284424559773903, −3.59848360718374381564888284336, −3.26321164020610104408044745777, −2.05315199560821415510065673884, −1.37132575934561351394632457756,
1.37132575934561351394632457756, 2.05315199560821415510065673884, 3.26321164020610104408044745777, 3.59848360718374381564888284336, 4.29928078103888284424559773903, 5.35919431520621974263615254112, 5.51763624678873988892869467209, 6.55840897780026521438484164195, 7.04964082746897053505966053312, 7.33008804031268611212863646820, 7.71201727717384931154224197875, 8.054107134123647440803206813455, 9.269489629824524181883533766445, 9.368509676824820831888843784883, 10.05806791301518125763005312962, 10.48053830642631756293273451889, 11.02037666658441009546729138770, 11.49802957873690594574763968636, 11.86674004305445302541064934117, 12.01571804133945693335686446554
