| L(s) = 1 | − 5·5-s − 2·7-s − 30·11-s + 4·13-s + 180·17-s − 56·19-s − 120·23-s − 210·29-s + 4·31-s + 10·35-s + 400·37-s − 240·41-s + 136·43-s + 120·47-s + 343·49-s − 60·53-s + 150·55-s + 450·59-s + 166·61-s − 20·65-s − 908·67-s − 2.04e3·71-s − 500·73-s + 60·77-s + 916·79-s + 1.14e3·83-s − 900·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.107·7-s − 0.822·11-s + 0.0853·13-s + 2.56·17-s − 0.676·19-s − 1.08·23-s − 1.34·29-s + 0.0231·31-s + 0.0482·35-s + 1.77·37-s − 0.914·41-s + 0.482·43-s + 0.372·47-s + 49-s − 0.155·53-s + 0.367·55-s + 0.992·59-s + 0.348·61-s − 0.0381·65-s − 1.65·67-s − 3.40·71-s − 0.801·73-s + 0.0888·77-s + 1.30·79-s + 1.50·83-s − 1.14·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1109970691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1109970691\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T - 339 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 30 T - 431 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T - 2181 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 90 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 28 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 120 T + 2233 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 210 T + 19711 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 4 T - 29775 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 200 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 240 T - 11321 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 136 T - 61011 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 120 T - 89423 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 30 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 450 T - 2879 T^{2} - 450 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 166 T - 199425 T^{2} - 166 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 908 T + 523701 T^{2} + 908 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 1020 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 250 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 916 T + 346017 T^{2} - 916 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 1140 T + 727813 T^{2} - 1140 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 420 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 1538 T + 1452771 T^{2} + 1538 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.577975142715556788500628192157, −8.713671906088203592311723777997, −8.368239522967900127809999200971, −7.86927649750114272249377465992, −7.80947908342359151375510984630, −7.25866805029369186443985096867, −7.09163031409123613086511297252, −6.17528960484236701451022299210, −5.99225580246452438874083777557, −5.52172674261808539951620271367, −5.34890602703545337268806780692, −4.52610248911436309000694764837, −4.24115095677107864646307847472, −3.60785198718358548573447097410, −3.42904024109421979011817073283, −2.53295497382072620116038362480, −2.48946374736915228736623506539, −1.37232588779270444710161745246, −1.17473545460130699301810812103, −0.07681495809115050831755858804,
0.07681495809115050831755858804, 1.17473545460130699301810812103, 1.37232588779270444710161745246, 2.48946374736915228736623506539, 2.53295497382072620116038362480, 3.42904024109421979011817073283, 3.60785198718358548573447097410, 4.24115095677107864646307847472, 4.52610248911436309000694764837, 5.34890602703545337268806780692, 5.52172674261808539951620271367, 5.99225580246452438874083777557, 6.17528960484236701451022299210, 7.09163031409123613086511297252, 7.25866805029369186443985096867, 7.80947908342359151375510984630, 7.86927649750114272249377465992, 8.368239522967900127809999200971, 8.713671906088203592311723777997, 9.577975142715556788500628192157
