| L(s) = 1 | + 3·2-s − 3·3-s + 8·4-s + 3·5-s − 9·6-s − 7·7-s + 45·8-s + 9·10-s + 15·11-s − 24·12-s − 128·13-s − 21·14-s − 9·15-s + 135·16-s − 84·17-s + 16·19-s + 24·20-s + 21·21-s + 45·22-s + 84·23-s − 135·24-s + 125·25-s − 384·26-s + 27·27-s − 56·28-s − 594·29-s − 27·30-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.577·3-s + 4-s + 0.268·5-s − 0.612·6-s − 0.377·7-s + 1.98·8-s + 0.284·10-s + 0.411·11-s − 0.577·12-s − 2.73·13-s − 0.400·14-s − 0.154·15-s + 2.10·16-s − 1.19·17-s + 0.193·19-s + 0.268·20-s + 0.218·21-s + 0.436·22-s + 0.761·23-s − 1.14·24-s + 25-s − 2.89·26-s + 0.192·27-s − 0.377·28-s − 3.80·29-s − 0.164·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.770138298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.770138298\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 + p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T - 116 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 15 T - 1106 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 64 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 84 T + 2143 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 16 T - 6603 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 84 T - 5111 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 297 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 253 T + 34218 T^{2} - 253 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 316 T + 49203 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 360 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 26 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T - 102923 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 363 T - 17108 T^{2} + 363 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 15 T - 205154 T^{2} - 15 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T - 213057 T^{2} - 118 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 370 T - 163863 T^{2} - 370 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 342 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 362 T - 257973 T^{2} + 362 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 467 T - 274950 T^{2} + 467 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 477 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 906 T + 115867 T^{2} + 906 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 503 T + p^{3} T^{2} )^{2} \) |
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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
−17.58386982955827734240686480395, −17.23074733756588066166355793207, −16.67477650348304783612437507985, −16.30416758363011446028043546281, −15.27106905125728587197844724311, −14.60107513759248829817160017074, −14.40025286648667157123636101921, −13.11785989273716789651633747066, −13.00932163019732403024923293547, −12.28229638658006112017284905068, −11.20719166378544860614006310948, −11.07506946322079037619428719899, −9.873668919387526237337095414518, −9.361730012047625925671515567963, −7.51666512192279009581180184974, −7.24478076920553501392747320947, −6.10912565339455725894963758727, −5.03511410905422182392571092911, −4.37886213680069290113882929938, −2.45162046213851339139222966109,
2.45162046213851339139222966109, 4.37886213680069290113882929938, 5.03511410905422182392571092911, 6.10912565339455725894963758727, 7.24478076920553501392747320947, 7.51666512192279009581180184974, 9.361730012047625925671515567963, 9.873668919387526237337095414518, 11.07506946322079037619428719899, 11.20719166378544860614006310948, 12.28229638658006112017284905068, 13.00932163019732403024923293547, 13.11785989273716789651633747066, 14.40025286648667157123636101921, 14.60107513759248829817160017074, 15.27106905125728587197844724311, 16.30416758363011446028043546281, 16.67477650348304783612437507985, 17.23074733756588066166355793207, 17.58386982955827734240686480395
