| L(s) = 1 | + 4·3-s + 3·4-s − 4·5-s + 8·9-s + 12·12-s − 16·15-s + 5·16-s − 4·17-s − 12·20-s + 11·25-s + 12·27-s + 2·29-s − 12·31-s + 24·36-s − 12·37-s − 32·45-s + 16·47-s + 20·48-s − 16·51-s − 18·53-s − 8·59-s − 48·60-s + 2·61-s + 3·64-s + 12·67-s − 12·68-s − 8·71-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 3/2·4-s − 1.78·5-s + 8/3·9-s + 3.46·12-s − 4.13·15-s + 5/4·16-s − 0.970·17-s − 2.68·20-s + 11/5·25-s + 2.30·27-s + 0.371·29-s − 2.15·31-s + 4·36-s − 1.97·37-s − 4.77·45-s + 2.33·47-s + 2.88·48-s − 2.24·51-s − 2.47·53-s − 1.04·59-s − 6.19·60-s + 0.256·61-s + 3/8·64-s + 1.46·67-s − 1.45·68-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.649449750\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.649449750\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + 12 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
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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
−12.55433940273585752199393775436, −12.44465047448325901546869438037, −12.07899341653380096266400353145, −11.14272400612681228924768427373, −10.96463272344082597549025839924, −10.68083735768657670479438398691, −9.704381549557454118493819669876, −9.021796044389018086431107144101, −8.814969308587277167997006841652, −8.190977528758344099280732312566, −7.78906274719293863198564271169, −7.40003925241022711062202436238, −6.92413612331218974735430151440, −6.47071869275487498402387649861, −5.23760976003029564033001210783, −4.33557436344700004903326951047, −3.53554322382929256253293570366, −3.37276573642555164478819359913, −2.53213800253828698995579105100, −1.89512979356304239687861555903,
1.89512979356304239687861555903, 2.53213800253828698995579105100, 3.37276573642555164478819359913, 3.53554322382929256253293570366, 4.33557436344700004903326951047, 5.23760976003029564033001210783, 6.47071869275487498402387649861, 6.92413612331218974735430151440, 7.40003925241022711062202436238, 7.78906274719293863198564271169, 8.190977528758344099280732312566, 8.814969308587277167997006841652, 9.021796044389018086431107144101, 9.704381549557454118493819669876, 10.68083735768657670479438398691, 10.96463272344082597549025839924, 11.14272400612681228924768427373, 12.07899341653380096266400353145, 12.44465047448325901546869438037, 12.55433940273585752199393775436
