| L(s) = 1 | − 2·2-s + 7·3-s + 8·4-s + 7·5-s − 14·6-s − 40·8-s + 27·9-s − 14·10-s + 5·11-s + 56·12-s + 28·13-s + 49·15-s + 80·16-s − 21·17-s − 54·18-s + 49·19-s + 56·20-s − 10·22-s + 159·23-s − 280·24-s + 125·25-s − 56·26-s + 224·27-s + 116·29-s − 98·30-s + 147·31-s − 320·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·3-s + 4-s + 0.626·5-s − 0.952·6-s − 1.76·8-s + 9-s − 0.442·10-s + 0.137·11-s + 1.34·12-s + 0.597·13-s + 0.843·15-s + 5/4·16-s − 0.299·17-s − 0.707·18-s + 0.591·19-s + 0.626·20-s − 0.0969·22-s + 1.44·23-s − 2.38·24-s + 25-s − 0.422·26-s + 1.59·27-s + 0.742·29-s − 0.596·30-s + 0.851·31-s − 1.76·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.204329091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.204329091\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T - p^{2} T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 7 T + 22 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T - 76 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T - 1306 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 21 T - 4472 T^{2} + 21 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 49 T - 4458 T^{2} - 49 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 159 T + 13114 T^{2} - 159 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 147 T - 8182 T^{2} - 147 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 219 T - 2692 T^{2} + 219 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 350 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 124 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 525 T + 171802 T^{2} - 525 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 303 T - 57068 T^{2} + 303 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 105 T - 194354 T^{2} + 105 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 413 T - 56412 T^{2} + 413 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 415 T - 128538 T^{2} + 415 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 432 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 1113 T + 849752 T^{2} + 1113 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 103 T - 482430 T^{2} - 103 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 1092 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 329 T - 596728 T^{2} + 329 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 882 T + p^{3} T^{2} )^{2} \) |
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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
−15.57618236274701173286316897747, −14.77966723689247174771296786909, −14.47954291492294004914385608361, −13.63368336898788013741006311956, −13.37551521284181927205018725931, −12.40958691899981297245741830856, −11.98057225477039222686900033933, −11.21910479427611558524128774496, −10.39845376101178218881039258488, −10.02328686617267129357396298576, −9.088848293227599325778248676361, −8.686210137971942493694318853106, −8.507602741994005857856066353656, −7.22913365859080400240285678282, −6.79962182325509169258085329470, −6.02964880306457286738376039210, −4.87482792599805007294665219114, −3.15495962223867153442379067193, −2.89647039959383912277523104768, −1.48202847095998175464269468530,
1.48202847095998175464269468530, 2.89647039959383912277523104768, 3.15495962223867153442379067193, 4.87482792599805007294665219114, 6.02964880306457286738376039210, 6.79962182325509169258085329470, 7.22913365859080400240285678282, 8.507602741994005857856066353656, 8.686210137971942493694318853106, 9.088848293227599325778248676361, 10.02328686617267129357396298576, 10.39845376101178218881039258488, 11.21910479427611558524128774496, 11.98057225477039222686900033933, 12.40958691899981297245741830856, 13.37551521284181927205018725931, 13.63368336898788013741006311956, 14.47954291492294004914385608361, 14.77966723689247174771296786909, 15.57618236274701173286316897747
