| L(s) = 1 | − 2-s − 4-s + 2·5-s + 3·8-s + 9-s − 2·10-s + 2·13-s − 16-s + 4·17-s − 18-s − 2·20-s + 3·25-s − 2·26-s − 4·29-s − 5·32-s − 4·34-s − 36-s + 12·37-s + 6·40-s − 12·41-s + 2·45-s − 14·49-s − 3·50-s − 2·52-s + 12·53-s + 4·58-s − 4·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.554·13-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.447·20-s + 3/5·25-s − 0.392·26-s − 0.742·29-s − 0.883·32-s − 0.685·34-s − 1/6·36-s + 1.97·37-s + 0.948·40-s − 1.87·41-s + 0.298·45-s − 2·49-s − 0.424·50-s − 0.277·52-s + 1.64·53-s + 0.525·58-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 608400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.482536669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.482536669\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−8.589432308568120532399394263407, −7.88541805670611931278793272721, −7.66093767427808108694408525270, −7.25526504565177991185560232138, −6.48654966834348628333399350819, −6.20235989153436615567604779531, −5.70617428080584220916058512582, −5.00649007098692076039682952910, −4.88823595164803979266926983864, −4.08370634935231621549895187340, −3.56799192480000330057521215275, −2.99305738219524609691519223861, −2.03833641567320312947705107247, −1.53827946418072491307494025034, −0.75387101601756041955604763488,
0.75387101601756041955604763488, 1.53827946418072491307494025034, 2.03833641567320312947705107247, 2.99305738219524609691519223861, 3.56799192480000330057521215275, 4.08370634935231621549895187340, 4.88823595164803979266926983864, 5.00649007098692076039682952910, 5.70617428080584220916058512582, 6.20235989153436615567604779531, 6.48654966834348628333399350819, 7.25526504565177991185560232138, 7.66093767427808108694408525270, 7.88541805670611931278793272721, 8.589432308568120532399394263407
