| L(s) = 1 | + 3·4-s − 9-s − 2·11-s + 5·16-s − 10·29-s − 2·31-s − 3·36-s + 12·41-s − 6·44-s + 13·49-s + 6·59-s − 14·61-s + 3·64-s + 24·79-s + 81-s − 16·89-s + 2·99-s − 6·101-s + 4·109-s − 30·116-s − 19·121-s − 6·124-s + 127-s + 131-s + 137-s + 139-s − 5·144-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 1/3·9-s − 0.603·11-s + 5/4·16-s − 1.85·29-s − 0.359·31-s − 1/2·36-s + 1.87·41-s − 0.904·44-s + 13/7·49-s + 0.781·59-s − 1.79·61-s + 3/8·64-s + 2.70·79-s + 1/9·81-s − 1.69·89-s + 0.201·99-s − 0.597·101-s + 0.383·109-s − 2.78·116-s − 1.72·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.416·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.666667415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.666667415\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 69 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−10.52434940516577168327865340994, −9.701763933474801837759326348018, −9.446759461177391178498549445805, −9.065418090989400501468343663135, −8.486204971751738744300218392421, −7.940697842689239057631078536823, −7.65759326503720371265935522019, −7.27344276489352481846552249237, −6.93422604418664686580821272631, −6.36917371395779687458389034173, −5.94396543644895071112042104231, −5.48291234248810505084750319914, −5.31681160649411163537451040130, −4.27103674759231294887935414309, −4.05005750577936122211108810791, −3.10336653665847457017880378324, −2.94085915972937908553601403264, −2.06797748971956842010687938389, −1.91696029519840575915877632473, −0.72386012556167078045287354427,
0.72386012556167078045287354427, 1.91696029519840575915877632473, 2.06797748971956842010687938389, 2.94085915972937908553601403264, 3.10336653665847457017880378324, 4.05005750577936122211108810791, 4.27103674759231294887935414309, 5.31681160649411163537451040130, 5.48291234248810505084750319914, 5.94396543644895071112042104231, 6.36917371395779687458389034173, 6.93422604418664686580821272631, 7.27344276489352481846552249237, 7.65759326503720371265935522019, 7.940697842689239057631078536823, 8.486204971751738744300218392421, 9.065418090989400501468343663135, 9.446759461177391178498549445805, 9.701763933474801837759326348018, 10.52434940516577168327865340994
