| L(s) = 1 | − 64·11-s − 64·16-s − 200·19-s − 100·29-s − 216·31-s − 44·41-s + 650·49-s + 1.00e3·59-s − 1.03e3·61-s − 824·71-s − 1.20e3·79-s − 300·89-s − 1.40e3·101-s + 1.10e3·109-s + 410·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.95e3·169-s + 173-s + 4.09e3·176-s + ⋯ |
| L(s) = 1 | − 1.75·11-s − 16-s − 2.41·19-s − 0.640·29-s − 1.25·31-s − 0.167·41-s + 1.89·49-s + 2.20·59-s − 2.17·61-s − 1.37·71-s − 1.70·79-s − 0.357·89-s − 1.38·101-s + 0.966·109-s + 0.308·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.34·169-s + 0.000439·173-s + 1.75·176-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5026066779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5026066779\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p^{2} T + p^{3} T^{2} )( 1 + p^{2} T + p^{3} T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 650 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2950 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9150 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 100 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 18250 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 50 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 108 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 30550 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 22 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 36350 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 56550 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 297750 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 500 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 518 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 585650 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 412 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 7150 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 600 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1064050 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 150 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1676350 T^{2} + p^{6} T^{4} \) |
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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
−12.66464164923169188111478288414, −11.48999159216930931950021529863, −10.91282225535868211705322114922, −10.53499610710502673366448436790, −10.42965859123453469199420682920, −9.555324731457588957912947919611, −9.085699558658319960723887401631, −8.383308826170656975080157514172, −8.362490277408615677388252496252, −7.32488124350269556007860738421, −7.21177933613361833415980915041, −6.43856895581418512275817067461, −5.69287594106284963827397012071, −5.43477343017973845487129109246, −4.37910657868203024511301877637, −4.29135622900123821162032689401, −3.15903330841775996080496884823, −2.37348008922852876239774577557, −1.91314019147042051215469236949, −0.27358447374549899257055219164,
0.27358447374549899257055219164, 1.91314019147042051215469236949, 2.37348008922852876239774577557, 3.15903330841775996080496884823, 4.29135622900123821162032689401, 4.37910657868203024511301877637, 5.43477343017973845487129109246, 5.69287594106284963827397012071, 6.43856895581418512275817067461, 7.21177933613361833415980915041, 7.32488124350269556007860738421, 8.362490277408615677388252496252, 8.383308826170656975080157514172, 9.085699558658319960723887401631, 9.555324731457588957912947919611, 10.42965859123453469199420682920, 10.53499610710502673366448436790, 10.91282225535868211705322114922, 11.48999159216930931950021529863, 12.66464164923169188111478288414
