| L(s) = 1 | − 16·4-s − 81·9-s + 432·11-s + 256·16-s − 1.76e3·19-s + 2.96e3·29-s + 1.67e4·31-s + 1.29e3·36-s − 1.95e4·41-s − 6.91e3·44-s − 2.40e3·49-s − 5.61e4·59-s − 7.77e4·61-s − 4.09e3·64-s − 4.12e4·71-s + 2.82e4·76-s + 1.99e5·79-s + 6.56e3·81-s − 7.27e4·89-s − 3.49e4·99-s + 3.69e5·101-s + 1.27e5·109-s − 4.74e4·116-s − 1.82e5·121-s − 2.67e5·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 1.07·11-s + 1/4·16-s − 1.12·19-s + 0.654·29-s + 3.12·31-s + 1/6·36-s − 1.81·41-s − 0.538·44-s − 1/7·49-s − 2.10·59-s − 2.67·61-s − 1/8·64-s − 0.971·71-s + 0.561·76-s + 3.58·79-s + 1/9·81-s − 0.973·89-s − 0.358·99-s + 3.60·101-s + 1.02·109-s − 0.327·116-s − 1.13·121-s − 1.56·124-s + 5.50e−6·127-s + 5.09e−6·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1102500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.526027658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.526027658\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{4} T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 216 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 253418 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 1144510 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 884 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 7728862 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 1482 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8360 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 116466118 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 9786 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 45290 p^{2} T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 34149986 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 118686886 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 28092 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 38866 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 2126743510 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 20628 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4146059086 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 99544 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 7505282422 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 36390 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 10921350430 T^{2} + p^{10} T^{4} \) |
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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
−9.211433126041990958408015172270, −9.075656973880108183425007232434, −8.468916438998643056892091228891, −8.346897353237490953036070411316, −7.78536465460013522213364755446, −7.43964068391740601324538375369, −6.54812730040931987597674076735, −6.37777010966495410606434202864, −6.33368499683026729038060166761, −5.58248338769560090595730521320, −4.82667214081940254692594484588, −4.65644010017826025905763986868, −4.30734053220898864157231200143, −3.66516338241220223454728099492, −2.94751200962044276835549287307, −2.94747374557849682972651990727, −1.76938231399989872068125491488, −1.69944604946496315646301623660, −0.62661402488640268972036140659, −0.57386559701590160890638792612,
0.57386559701590160890638792612, 0.62661402488640268972036140659, 1.69944604946496315646301623660, 1.76938231399989872068125491488, 2.94747374557849682972651990727, 2.94751200962044276835549287307, 3.66516338241220223454728099492, 4.30734053220898864157231200143, 4.65644010017826025905763986868, 4.82667214081940254692594484588, 5.58248338769560090595730521320, 6.33368499683026729038060166761, 6.37777010966495410606434202864, 6.54812730040931987597674076735, 7.43964068391740601324538375369, 7.78536465460013522213364755446, 8.346897353237490953036070411316, 8.468916438998643056892091228891, 9.075656973880108183425007232434, 9.211433126041990958408015172270
