| L(s) = 1 | − 4-s − 9-s − 12·11-s + 16-s + 2·19-s + 16·29-s − 16·31-s + 36-s − 8·41-s + 12·44-s + 10·49-s + 8·59-s + 4·61-s − 64-s − 2·76-s − 16·79-s + 81-s + 8·89-s + 12·99-s − 20·101-s + 28·109-s − 16·116-s + 86·121-s + 16·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s − 3.61·11-s + 1/4·16-s + 0.458·19-s + 2.97·29-s − 2.87·31-s + 1/6·36-s − 1.24·41-s + 1.80·44-s + 10/7·49-s + 1.04·59-s + 0.512·61-s − 1/8·64-s − 0.229·76-s − 1.80·79-s + 1/9·81-s + 0.847·89-s + 1.20·99-s − 1.99·101-s + 2.68·109-s − 1.48·116-s + 7.81·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8122500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4532424361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4532424361\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.748405958471388080676542011185, −8.514363960329833016188459269758, −8.195120690690702149987255975855, −8.070432006248144779169048415832, −7.37365270402183415445398658467, −7.10384103306805227964925950604, −7.07434882209134449263965328751, −5.95815623256053133809195794119, −5.92986547221285387275910221698, −5.30420021914845587654280960194, −5.26721247081828854607955720388, −4.67999193834739027564263963275, −4.56862626690688092125252795506, −3.57599299436849554045616775686, −3.40033153489839189689044095054, −2.75102892443070551879861934766, −2.48236430737732112372111426490, −2.03729841766325524846121265926, −1.03826223807395454268976341650, −0.24166529714376774706309971269,
0.24166529714376774706309971269, 1.03826223807395454268976341650, 2.03729841766325524846121265926, 2.48236430737732112372111426490, 2.75102892443070551879861934766, 3.40033153489839189689044095054, 3.57599299436849554045616775686, 4.56862626690688092125252795506, 4.67999193834739027564263963275, 5.26721247081828854607955720388, 5.30420021914845587654280960194, 5.92986547221285387275910221698, 5.95815623256053133809195794119, 7.07434882209134449263965328751, 7.10384103306805227964925950604, 7.37365270402183415445398658467, 8.070432006248144779169048415832, 8.195120690690702149987255975855, 8.514363960329833016188459269758, 8.748405958471388080676542011185
