| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·9-s + 2·10-s − 4·13-s + 16-s + 8·17-s + 2·18-s + 2·20-s + 3·25-s − 4·26-s − 8·29-s + 32-s + 8·34-s + 2·36-s + 2·37-s + 2·40-s + 12·41-s + 4·45-s + 6·49-s + 3·50-s − 4·52-s − 8·58-s + 64-s − 8·65-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 2/3·9-s + 0.632·10-s − 1.10·13-s + 1/4·16-s + 1.94·17-s + 0.471·18-s + 0.447·20-s + 3/5·25-s − 0.784·26-s − 1.48·29-s + 0.176·32-s + 1.37·34-s + 1/3·36-s + 0.328·37-s + 0.316·40-s + 1.87·41-s + 0.596·45-s + 6/7·49-s + 0.424·50-s − 0.554·52-s − 1.05·58-s + 1/8·64-s − 0.992·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.556794423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.556794423\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−7.80021022775430441531408809226, −7.58585527266948775876736657933, −7.20552315909597070533843902842, −6.78427025973501634097775334182, −6.15946106985185681748568436970, −5.67352503947890013636487329766, −5.51832655236226650629974468909, −5.04167159885489444977418212712, −4.37479970783640735547939347351, −4.04847985564756178140577258055, −3.35555804325760842479407023818, −2.83908832562660018083042083258, −2.26278253704023129981768029003, −1.66833694381427609691398347883, −0.905375457934824433129038326127,
0.905375457934824433129038326127, 1.66833694381427609691398347883, 2.26278253704023129981768029003, 2.83908832562660018083042083258, 3.35555804325760842479407023818, 4.04847985564756178140577258055, 4.37479970783640735547939347351, 5.04167159885489444977418212712, 5.51832655236226650629974468909, 5.67352503947890013636487329766, 6.15946106985185681748568436970, 6.78427025973501634097775334182, 7.20552315909597070533843902842, 7.58585527266948775876736657933, 7.80021022775430441531408809226
