| L(s) = 1 | + 4·5-s − 9-s + 8·11-s + 4·19-s + 11·25-s − 16·29-s + 16·31-s + 4·41-s − 4·45-s + 14·49-s + 32·55-s + 16·59-s − 4·61-s − 8·79-s + 81-s + 36·89-s + 16·95-s − 8·99-s − 28·109-s + 26·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1/3·9-s + 2.41·11-s + 0.917·19-s + 11/5·25-s − 2.97·29-s + 2.87·31-s + 0.624·41-s − 0.596·45-s + 2·49-s + 4.31·55-s + 2.08·59-s − 0.512·61-s − 0.900·79-s + 1/9·81-s + 3.81·89-s + 1.64·95-s − 0.804·99-s − 2.68·109-s + 2.36·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.091416458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.091416458\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−9.376593022453164231774860135119, −9.097988578398848108001284634355, −8.831889786338374693836291603307, −8.437842115809570456234223064393, −7.71882914218403255865438548019, −7.44650196409547714647423629253, −6.78091824710526643708146965853, −6.66607540982780313614421729813, −6.12606259622556967364825863661, −5.95765379326392013067487702741, −5.33493737452325880875416042331, −5.26030810621300690662355796819, −4.34687693852436240339053121936, −4.13158150538652941484931820674, −3.52633750517280425914967349299, −3.05955769792177528044634893145, −2.25502858019106743975962276257, −2.09894175507390421133175312001, −1.14829188857339280882400961813, −1.03507678188469915131560765714,
1.03507678188469915131560765714, 1.14829188857339280882400961813, 2.09894175507390421133175312001, 2.25502858019106743975962276257, 3.05955769792177528044634893145, 3.52633750517280425914967349299, 4.13158150538652941484931820674, 4.34687693852436240339053121936, 5.26030810621300690662355796819, 5.33493737452325880875416042331, 5.95765379326392013067487702741, 6.12606259622556967364825863661, 6.66607540982780313614421729813, 6.78091824710526643708146965853, 7.44650196409547714647423629253, 7.71882914218403255865438548019, 8.437842115809570456234223064393, 8.831889786338374693836291603307, 9.097988578398848108001284634355, 9.376593022453164231774860135119
