| L(s) = 1 | + 2·2-s + 3·4-s + 5-s + 4·8-s − 2·9-s + 2·10-s + 5·16-s − 4·18-s + 3·20-s + 2·29-s + 6·32-s − 6·36-s − 2·37-s + 4·40-s − 2·45-s + 2·49-s + 4·58-s + 2·61-s + 7·64-s − 8·72-s − 2·73-s − 4·74-s + 5·80-s + 3·81-s − 4·90-s − 4·97-s + 4·98-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 5-s + 4·8-s − 2·9-s + 2·10-s + 5·16-s − 4·18-s + 3·20-s + 2·29-s + 6·32-s − 6·36-s − 2·37-s + 4·40-s − 2·45-s + 2·49-s + 4·58-s + 2·61-s + 7·64-s − 8·72-s − 2·73-s − 4·74-s + 5·80-s + 3·81-s − 4·90-s − 4·97-s + 4·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11424400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11424400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(6.361604111\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.361604111\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$ | \( ( 1 + T )^{4} \) |
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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
−8.681485745234346018779047302873, −8.663472449575401113603482016642, −8.118321905289042751165889697195, −7.917423618796417269441643969804, −7.05502869548159628382395535097, −7.03084904299394981646434362147, −6.55266292757027553296092534227, −6.24172002732736456763090227308, −5.74641999505331108263344870248, −5.49502191801998934046410731599, −5.35909502598780485538030469732, −4.97816530325951853797434639285, −4.24678462749496717848847112793, −4.08217057206379429356308869670, −3.42086498912201248122101115726, −3.02281811361872055438906151895, −2.56429452904242763744076667974, −2.46514215312052045295560333167, −1.76531196172674392041617007576, −1.13592755646245345729603380383,
1.13592755646245345729603380383, 1.76531196172674392041617007576, 2.46514215312052045295560333167, 2.56429452904242763744076667974, 3.02281811361872055438906151895, 3.42086498912201248122101115726, 4.08217057206379429356308869670, 4.24678462749496717848847112793, 4.97816530325951853797434639285, 5.35909502598780485538030469732, 5.49502191801998934046410731599, 5.74641999505331108263344870248, 6.24172002732736456763090227308, 6.55266292757027553296092534227, 7.03084904299394981646434362147, 7.05502869548159628382395535097, 7.917423618796417269441643969804, 8.118321905289042751165889697195, 8.663472449575401113603482016642, 8.681485745234346018779047302873
