| L(s) = 1 | − 4·5-s + 5·7-s − 6·11-s + 10·13-s + 2·17-s + 19-s + 6·23-s + 5·25-s − 3·31-s − 20·35-s − 3·37-s + 12·41-s − 10·43-s + 4·47-s + 18·49-s − 6·53-s + 24·55-s + 6·59-s + 2·61-s − 40·65-s + 7·67-s + 32·71-s + 3·73-s − 30·77-s + 11·79-s + 24·83-s − 8·85-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1.88·7-s − 1.80·11-s + 2.77·13-s + 0.485·17-s + 0.229·19-s + 1.25·23-s + 25-s − 0.538·31-s − 3.38·35-s − 0.493·37-s + 1.87·41-s − 1.52·43-s + 0.583·47-s + 18/7·49-s − 0.824·53-s + 3.23·55-s + 0.781·59-s + 0.256·61-s − 4.96·65-s + 0.855·67-s + 3.79·71-s + 0.351·73-s − 3.41·77-s + 1.23·79-s + 2.63·83-s − 0.867·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.563481723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.563481723\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−9.100487577212695413056994277020, −8.824755302780132110514277554088, −8.242218281435183352536760716263, −8.160177517000363723459361580824, −7.85330466796297915004207747358, −7.77478051231418797909575514521, −7.10239417724314872657937766386, −6.75483897660005135556884315620, −6.13378661811690869059310885874, −5.61524747117450991024449085357, −5.14721778978883171253185514551, −5.05306819502997888894083376791, −4.45467933550838778263732910673, −3.89955087739940349222494274180, −3.52980311674354794443760840815, −3.38233598413454419126362916744, −2.42731402077276535099807093052, −1.95279464218411590062008928367, −1.04448515017775994013852254602, −0.73383834323773110172313976429,
0.73383834323773110172313976429, 1.04448515017775994013852254602, 1.95279464218411590062008928367, 2.42731402077276535099807093052, 3.38233598413454419126362916744, 3.52980311674354794443760840815, 3.89955087739940349222494274180, 4.45467933550838778263732910673, 5.05306819502997888894083376791, 5.14721778978883171253185514551, 5.61524747117450991024449085357, 6.13378661811690869059310885874, 6.75483897660005135556884315620, 7.10239417724314872657937766386, 7.77478051231418797909575514521, 7.85330466796297915004207747358, 8.160177517000363723459361580824, 8.242218281435183352536760716263, 8.824755302780132110514277554088, 9.100487577212695413056994277020
