| L(s) = 1 | + 4-s + 3·5-s + 4·7-s − 3·11-s + 3·13-s − 3·16-s + 3·17-s − 9·19-s + 3·20-s − 9·23-s + 5·25-s + 4·28-s + 9·29-s + 12·35-s − 7·37-s + 3·41-s − 43-s − 3·44-s + 9·49-s + 3·52-s − 15·53-s − 9·55-s − 7·64-s + 9·65-s − 8·67-s + 3·68-s − 9·73-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.34·5-s + 1.51·7-s − 0.904·11-s + 0.832·13-s − 3/4·16-s + 0.727·17-s − 2.06·19-s + 0.670·20-s − 1.87·23-s + 25-s + 0.755·28-s + 1.67·29-s + 2.02·35-s − 1.15·37-s + 0.468·41-s − 0.152·43-s − 0.452·44-s + 9/7·49-s + 0.416·52-s − 2.06·53-s − 1.21·55-s − 7/8·64-s + 1.11·65-s − 0.977·67-s + 0.363·68-s − 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.936180580\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.936180580\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T + 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 15 T + 128 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 9 T + 100 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 15 T + 142 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 3 T - 80 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 3 T + 100 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−12.86706278285562702997062033078, −12.11376097525866690997451635786, −12.04191054325712859493301479954, −11.13167810783007110224673052767, −10.76065097039486489265060144778, −10.53119838694185963234949928565, −10.00279037089535379135670466250, −9.396968915543597441072449387229, −8.487508167254132979072366112366, −8.444276237637559525034583011494, −7.88773791311141527155313763300, −7.12425691836572134446665467770, −6.33134503448031498332437570336, −6.10379108144574420531505957857, −5.46118461214875833774081558645, −4.68673761222492747515077923309, −4.29954279178598716552846928541, −3.00861810588390214752082583317, −2.05497299292172316669943338762, −1.75705901659726654071396250373,
1.75705901659726654071396250373, 2.05497299292172316669943338762, 3.00861810588390214752082583317, 4.29954279178598716552846928541, 4.68673761222492747515077923309, 5.46118461214875833774081558645, 6.10379108144574420531505957857, 6.33134503448031498332437570336, 7.12425691836572134446665467770, 7.88773791311141527155313763300, 8.444276237637559525034583011494, 8.487508167254132979072366112366, 9.396968915543597441072449387229, 10.00279037089535379135670466250, 10.53119838694185963234949928565, 10.76065097039486489265060144778, 11.13167810783007110224673052767, 12.04191054325712859493301479954, 12.11376097525866690997451635786, 12.86706278285562702997062033078
