| L(s) = 1 | − 16·4-s + 768·11-s + 256·16-s + 1.32e3·19-s − 192·29-s − 9.12e3·31-s − 1.34e4·41-s − 1.22e4·44-s + 1.17e4·49-s + 2.61e4·59-s + 8.55e4·61-s − 4.09e3·64-s + 1.29e5·71-s − 2.12e4·76-s − 5.61e4·79-s + 1.63e5·89-s + 3.57e5·101-s + 4.62e5·109-s + 3.07e3·116-s + 1.20e5·121-s + 1.46e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.91·11-s + 1/4·16-s + 0.843·19-s − 0.0423·29-s − 1.70·31-s − 1.24·41-s − 0.956·44-s + 0.696·49-s + 0.976·59-s + 2.94·61-s − 1/8·64-s + 3.03·71-s − 0.421·76-s − 1.01·79-s + 2.18·89-s + 3.48·101-s + 3.72·109-s + 0.0211·116-s + 0.746·121-s + 0.852·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.818442346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.818442346\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 11710 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 384 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 631030 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2507938 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 664 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 1872914 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 96 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4564 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 105071110 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6720 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 72840502 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90050014 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 775924810 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 13056 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 42782 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 1356587878 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 64512 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3863567086 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 28076 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3464870662 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 81792 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 16278396670 T^{2} + p^{10} T^{4} \) |
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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
−10.17032535483729759996765050339, −10.16490397046606604756889362883, −9.574956874725974396088810053022, −9.124578571066427613644576643936, −8.676441893540621789367844496240, −8.573932417044861998537832914963, −7.56145422401294351932073296518, −7.45449009513509772158924320010, −6.60592916217333221512572463878, −6.56057659951772148883116520480, −5.72597110602211587191814040087, −5.27179300142151465265446504070, −4.82873368931448336159888392019, −4.05447922885184289198486501316, −3.53894523194962432745134551621, −3.48491408541956880535222258337, −2.22754804753113614030135066722, −1.80362342114274887238597520393, −0.908637679731059634143987254474, −0.60743799720323446139953730640,
0.60743799720323446139953730640, 0.908637679731059634143987254474, 1.80362342114274887238597520393, 2.22754804753113614030135066722, 3.48491408541956880535222258337, 3.53894523194962432745134551621, 4.05447922885184289198486501316, 4.82873368931448336159888392019, 5.27179300142151465265446504070, 5.72597110602211587191814040087, 6.56057659951772148883116520480, 6.60592916217333221512572463878, 7.45449009513509772158924320010, 7.56145422401294351932073296518, 8.573932417044861998537832914963, 8.676441893540621789367844496240, 9.124578571066427613644576643936, 9.574956874725974396088810053022, 10.16490397046606604756889362883, 10.17032535483729759996765050339
