| L(s) = 1 | + 2·2-s − 4-s − 8·8-s + 9-s + 8·13-s − 7·16-s + 2·18-s − 2·19-s − 6·25-s + 16·26-s + 14·32-s − 36-s − 4·38-s + 7·43-s + 16·47-s + 10·49-s − 12·50-s − 8·52-s + 2·59-s + 35·64-s − 6·67-s − 8·72-s + 2·76-s + 81-s − 12·83-s + 14·86-s + 24·89-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s − 2.82·8-s + 1/3·9-s + 2.21·13-s − 7/4·16-s + 0.471·18-s − 0.458·19-s − 6/5·25-s + 3.13·26-s + 2.47·32-s − 1/6·36-s − 0.648·38-s + 1.06·43-s + 2.33·47-s + 10/7·49-s − 1.69·50-s − 1.10·52-s + 0.260·59-s + 35/8·64-s − 0.733·67-s − 0.942·72-s + 0.229·76-s + 1/9·81-s − 1.31·83-s + 1.50·86-s + 2.54·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111843 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111843 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.188668786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.188668786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_2$ | \( 1 + p T^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
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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
−9.342645758600720603857631980131, −8.934608556960826440897972022847, −8.672019587496576355288604211159, −8.137223190130620346244021402948, −7.51313590134417345340192795702, −6.72998257582812924553674393985, −6.07412166431298571651843046137, −5.77636885232380532870072926026, −5.52282777440245414682367955061, −4.51803479233245618213286824591, −4.22422488276041012287773462549, −3.76558688285872711998255244692, −3.30695030192831012719784700008, −2.33169475936642155869684830244, −0.911293197169709479379781494088,
0.911293197169709479379781494088, 2.33169475936642155869684830244, 3.30695030192831012719784700008, 3.76558688285872711998255244692, 4.22422488276041012287773462549, 4.51803479233245618213286824591, 5.52282777440245414682367955061, 5.77636885232380532870072926026, 6.07412166431298571651843046137, 6.72998257582812924553674393985, 7.51313590134417345340192795702, 8.137223190130620346244021402948, 8.672019587496576355288604211159, 8.934608556960826440897972022847, 9.342645758600720603857631980131
