| L(s) = 1 | − 4·5-s − 3·7-s − 28·11-s + 11·13-s + 88·17-s + 58·19-s − 172·23-s + 125·25-s − 192·29-s − 116·31-s + 12·35-s − 138·37-s − 384·41-s − 328·43-s − 156·47-s + 343·49-s − 784·53-s + 112·55-s − 412·59-s + 425·61-s − 44·65-s − 257·67-s − 2.00e3·71-s − 718·73-s + 84·77-s − 877·79-s + 328·83-s + ⋯ |
| L(s) = 1 | − 0.357·5-s − 0.161·7-s − 0.767·11-s + 0.234·13-s + 1.25·17-s + 0.700·19-s − 1.55·23-s + 25-s − 1.22·29-s − 0.672·31-s + 0.0579·35-s − 0.613·37-s − 1.46·41-s − 1.16·43-s − 0.484·47-s + 49-s − 2.03·53-s + 0.274·55-s − 0.909·59-s + 0.892·61-s − 0.0839·65-s − 0.468·67-s − 3.34·71-s − 1.15·73-s + 0.124·77-s − 1.24·79-s + 0.433·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1142364922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1142364922\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T - 109 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 3 T - 334 T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 28 T - 547 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 11 T - 2076 T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 44 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 29 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 172 T + 17417 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 192 T + 12475 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 116 T - 16335 T^{2} + 116 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 69 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 384 T + 78535 T^{2} + 384 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 328 T + 28077 T^{2} + 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 156 T - 79487 T^{2} + 156 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 392 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 412 T - 35635 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 425 T - 46356 T^{2} - 425 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 257 T - 234714 T^{2} + 257 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1000 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 359 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 877 T + 276090 T^{2} + 877 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 328 T - 464203 T^{2} - 328 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1572 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1483 T + 1286616 T^{2} - 1483 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.47362736808719738266358984860, −9.820218308157106674603338489164, −9.803586214842984923001502768036, −8.870697933595930938982494850057, −8.766220573377818803061991285253, −8.051510484432067680178575118528, −7.83710455216326123862722090300, −7.16769827192836269803740190009, −7.15038691480305695229992723516, −6.15767905482969988938885064332, −5.94739116337177163680974560944, −5.22674783140479214483519109481, −5.09960772482325228198984868192, −4.22163461386539498193456493056, −3.78612797171542236881154582129, −3.02922855041636864156497544124, −2.95882826096766963076725284060, −1.67001074949305629272307819952, −1.46505242564859219650657255533, −0.095178750804725917289445888137,
0.095178750804725917289445888137, 1.46505242564859219650657255533, 1.67001074949305629272307819952, 2.95882826096766963076725284060, 3.02922855041636864156497544124, 3.78612797171542236881154582129, 4.22163461386539498193456493056, 5.09960772482325228198984868192, 5.22674783140479214483519109481, 5.94739116337177163680974560944, 6.15767905482969988938885064332, 7.15038691480305695229992723516, 7.16769827192836269803740190009, 7.83710455216326123862722090300, 8.051510484432067680178575118528, 8.766220573377818803061991285253, 8.870697933595930938982494850057, 9.803586214842984923001502768036, 9.820218308157106674603338489164, 10.47362736808719738266358984860
