| L(s) = 1 | + 6·2-s + 9·3-s + 32·4-s + 6·5-s + 54·6-s + 360·8-s + 36·10-s + 564·11-s + 288·12-s − 1.27e3·13-s + 54·15-s + 2.16e3·16-s + 882·17-s − 556·19-s + 192·20-s + 3.38e3·22-s + 840·23-s + 3.24e3·24-s + 3.12e3·25-s − 7.65e3·26-s − 729·27-s + 9.27e3·29-s + 324·30-s + 4.40e3·31-s + 1.15e4·32-s + 5.07e3·33-s + 5.29e3·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.577·3-s + 4-s + 0.107·5-s + 0.612·6-s + 1.98·8-s + 0.113·10-s + 1.40·11-s + 0.577·12-s − 2.09·13-s + 0.0619·15-s + 2.10·16-s + 0.740·17-s − 0.353·19-s + 0.107·20-s + 1.49·22-s + 0.331·23-s + 1.14·24-s + 25-s − 2.22·26-s − 0.192·27-s + 2.04·29-s + 0.0657·30-s + 0.822·31-s + 1.98·32-s + 0.811·33-s + 0.785·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(9.725218301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.725218301\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p^{2} T + p^{4} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 p T + p^{2} T^{2} - 3 p^{6} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 6 T - 3089 T^{2} - 6 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 564 T + 157045 T^{2} - 564 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 638 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 882 T - 641933 T^{2} - 882 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 556 T - 2166963 T^{2} + 556 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 840 T - 5730743 T^{2} - 840 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4638 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 4400 T - 9269151 T^{2} - 4400 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2410 T - 63535857 T^{2} - 2410 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6870 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 9644 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 18672 T + 119298577 T^{2} + 18672 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 33750 T + 720867007 T^{2} + 33750 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 18084 T - 387893243 T^{2} + 18084 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 39758 T + 736102263 T^{2} - 39758 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 23068 T - 817992483 T^{2} - 23068 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 4248 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 41110 T - 383039493 T^{2} + 41110 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 21920 T - 2596569999 T^{2} + 21920 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 82452 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 94086 T + 3268115947 T^{2} + 94086 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 49442 T + p^{5} T^{2} )^{2} \) |
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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
−12.38637819544311159614023452449, −12.28826732778739372803365432483, −11.25285101700226641835806056872, −11.24593080364823343096891235868, −10.21785686471469451606614252147, −9.981657680043451480668025341524, −9.481217303539796639845125470487, −8.678594054529886608447847133321, −8.027834582003234374403179407300, −7.58075746044849668533204025950, −6.87279101375559204176294763135, −6.66211101970061420107310415456, −5.74804721541467509231254953905, −4.98522971172979653374841502881, −4.44644668323964549927164612433, −4.14357440964650173192268079544, −2.86478019883325987628196304306, −2.74363700874828018647191067535, −1.60958163553911845486059303783, −0.926515544954513050627977026271,
0.926515544954513050627977026271, 1.60958163553911845486059303783, 2.74363700874828018647191067535, 2.86478019883325987628196304306, 4.14357440964650173192268079544, 4.44644668323964549927164612433, 4.98522971172979653374841502881, 5.74804721541467509231254953905, 6.66211101970061420107310415456, 6.87279101375559204176294763135, 7.58075746044849668533204025950, 8.027834582003234374403179407300, 8.678594054529886608447847133321, 9.481217303539796639845125470487, 9.981657680043451480668025341524, 10.21785686471469451606614252147, 11.24593080364823343096891235868, 11.25285101700226641835806056872, 12.28826732778739372803365432483, 12.38637819544311159614023452449
