| L(s) = 1 | − 24·3-s + 333·9-s − 1.29e3·11-s − 484·13-s − 2.59e3·23-s − 86·25-s − 2.16e3·27-s + 3.11e4·33-s − 2.41e4·37-s + 1.16e4·39-s − 2.59e4·47-s + 1.53e3·49-s − 1.68e4·59-s − 5.15e4·61-s + 6.22e4·69-s − 1.11e5·71-s + 5.20e4·73-s + 2.06e3·75-s − 2.90e4·81-s + 1.56e5·83-s + 2.06e5·97-s − 4.31e5·99-s − 4.27e4·107-s + 1.17e5·109-s + 5.78e5·111-s − 1.61e5·117-s + 9.37e5·121-s + ⋯ |
| L(s) = 1 | − 1.53·3-s + 1.37·9-s − 3.22·11-s − 0.794·13-s − 1.02·23-s − 0.0275·25-s − 0.570·27-s + 4.97·33-s − 2.89·37-s + 1.22·39-s − 1.71·47-s + 0.0915·49-s − 0.630·59-s − 1.77·61-s + 1.57·69-s − 2.62·71-s + 1.14·73-s + 0.0423·75-s − 0.492·81-s + 2.49·83-s + 2.22·97-s − 4.42·99-s − 0.361·107-s + 0.949·109-s + 4.45·111-s − 1.08·117-s + 5.82·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 8 p T + p^{5} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 86 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 1538 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 648 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 242 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2738338 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 3380474 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 1296 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 37062298 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 45678866 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 12058 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 12512146 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 33190390 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12960 T + p^{5} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 108251530 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8424 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 25762 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2596035290 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 55728 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 26026 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 5786874674 T^{2} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 78408 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4075828594 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 103090 T + p^{5} T^{2} )^{2} \) |
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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
−14.33346948766056258267794338378, −13.54271200950747265327486131956, −13.14954214376105171616461895569, −12.42618700674898880868453116021, −12.16803681447798024406692650556, −11.40817616591633440147461784456, −10.55007337196433036720147916188, −10.44614794694244681362836589761, −10.00052018747835621863036919697, −8.822024221438508958533420036765, −7.79357343495356938918471518907, −7.59277545207133931989911948695, −6.56196593767845321582486799682, −5.73618633400204266377797123485, −5.06926050912115599428135938119, −4.82243143516014601518418114818, −3.15372104839205746772019009746, −2.01506447221583229123664631016, 0, 0,
2.01506447221583229123664631016, 3.15372104839205746772019009746, 4.82243143516014601518418114818, 5.06926050912115599428135938119, 5.73618633400204266377797123485, 6.56196593767845321582486799682, 7.59277545207133931989911948695, 7.79357343495356938918471518907, 8.822024221438508958533420036765, 10.00052018747835621863036919697, 10.44614794694244681362836589761, 10.55007337196433036720147916188, 11.40817616591633440147461784456, 12.16803681447798024406692650556, 12.42618700674898880868453116021, 13.14954214376105171616461895569, 13.54271200950747265327486131956, 14.33346948766056258267794338378
