| L(s) = 1 | − 3·5-s − 5·9-s − 2·11-s + 16·19-s + 4·25-s + 10·31-s + 15·45-s − 10·49-s + 6·55-s + 6·59-s − 8·61-s + 30·71-s + 4·79-s + 16·81-s − 18·89-s − 48·95-s + 10·99-s + 36·101-s + 4·109-s + 3·121-s + 3·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 5/3·9-s − 0.603·11-s + 3.67·19-s + 4/5·25-s + 1.79·31-s + 2.23·45-s − 1.42·49-s + 0.809·55-s + 0.781·59-s − 1.02·61-s + 3.56·71-s + 0.450·79-s + 16/9·81-s − 1.90·89-s − 4.92·95-s + 1.00·99-s + 3.58·101-s + 0.383·109-s + 3/11·121-s + 0.268·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8686529049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8686529049\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−9.828597367466048724363215268486, −9.813753565180213813561671136388, −9.062308306545389100350755701537, −8.323465712127201941198713831607, −8.132665807481574488057539990847, −7.60852684015633966980596452192, −7.19880969537039357464113305235, −6.37561722404072890002203233082, −5.71468598548784473589814335416, −5.11227132195889762323685903220, −4.78230154364361768976603326542, −3.54794291769774557092878092602, −3.27381985666804460223319703873, −2.62441060321725601358270028596, −0.809727221508988329410874951537,
0.809727221508988329410874951537, 2.62441060321725601358270028596, 3.27381985666804460223319703873, 3.54794291769774557092878092602, 4.78230154364361768976603326542, 5.11227132195889762323685903220, 5.71468598548784473589814335416, 6.37561722404072890002203233082, 7.19880969537039357464113305235, 7.60852684015633966980596452192, 8.132665807481574488057539990847, 8.323465712127201941198713831607, 9.062308306545389100350755701537, 9.813753565180213813561671136388, 9.828597367466048724363215268486
