L(s) = 1 | + 3-s − 2·9-s − 2·11-s − 8·13-s − 6·23-s − 25-s − 5·27-s − 2·33-s − 2·37-s − 8·39-s − 10·49-s + 6·59-s − 8·61-s − 6·69-s + 30·71-s − 8·73-s − 75-s + 81-s + 12·83-s − 14·97-s + 4·99-s + 12·107-s + 4·109-s − 2·111-s + 16·117-s + 3·121-s + 127-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.603·11-s − 2.21·13-s − 1.25·23-s − 1/5·25-s − 0.962·27-s − 0.348·33-s − 0.328·37-s − 1.28·39-s − 1.42·49-s + 0.781·59-s − 1.02·61-s − 0.722·69-s + 3.56·71-s − 0.936·73-s − 0.115·75-s + 1/9·81-s + 1.31·83-s − 1.42·97-s + 0.402·99-s + 1.16·107-s + 0.383·109-s − 0.189·111-s + 1.47·117-s + 3/11·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−9.726921937470959380626935593622, −9.205754524823445845175580209741, −8.408943998062523444280533863335, −8.132665807481574488057539990847, −7.60852684015633966980596452192, −7.22282278046792198667036847650, −6.47645013269452670867443390425, −5.90441514720049446274076604978, −5.11227132195889762323685903220, −4.92614337906569626690399921734, −3.98508995562062136618490430533, −3.27381985666804460223319703873, −2.49109929152055119200232183305, −2.07797879851041904182307638354, 0,
2.07797879851041904182307638354, 2.49109929152055119200232183305, 3.27381985666804460223319703873, 3.98508995562062136618490430533, 4.92614337906569626690399921734, 5.11227132195889762323685903220, 5.90441514720049446274076604978, 6.47645013269452670867443390425, 7.22282278046792198667036847650, 7.60852684015633966980596452192, 8.132665807481574488057539990847, 8.408943998062523444280533863335, 9.205754524823445845175580209741, 9.726921937470959380626935593622