| L(s) = 1 | + 2-s + 4-s + 8-s − 6·9-s + 8·11-s + 16-s − 12·17-s − 6·18-s + 8·22-s − 6·25-s + 32-s − 12·34-s − 6·36-s − 12·41-s + 8·43-s + 8·44-s + 49-s − 6·50-s + 16·59-s + 64-s + 8·67-s − 12·68-s − 6·72-s + 4·73-s + 27·81-s − 12·82-s + 8·86-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2·9-s + 2.41·11-s + 1/4·16-s − 2.91·17-s − 1.41·18-s + 1.70·22-s − 6/5·25-s + 0.176·32-s − 2.05·34-s − 36-s − 1.87·41-s + 1.21·43-s + 1.20·44-s + 1/7·49-s − 0.848·50-s + 2.08·59-s + 1/8·64-s + 0.977·67-s − 1.45·68-s − 0.707·72-s + 0.468·73-s + 3·81-s − 1.32·82-s + 0.862·86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1059968 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1059968 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.334405926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.334405926\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.377446654325248117350946284188, −7.49268064446268010230997600659, −7.08827055361374002488782088399, −6.39672270838488175514488859499, −6.34504575623978833805361946068, −6.19217622188912966190136871625, −5.21535057877927416914375242112, −5.10964147164568756325362418731, −4.29525367619910026638334030739, −3.84393814699741018049368924058, −3.67764363944170895713490289621, −2.83079975125625245447577863958, −2.18713180950649411226712364496, −1.88517885618852680356987113967, −0.59318722140774052200839944464,
0.59318722140774052200839944464, 1.88517885618852680356987113967, 2.18713180950649411226712364496, 2.83079975125625245447577863958, 3.67764363944170895713490289621, 3.84393814699741018049368924058, 4.29525367619910026638334030739, 5.10964147164568756325362418731, 5.21535057877927416914375242112, 6.19217622188912966190136871625, 6.34504575623978833805361946068, 6.39672270838488175514488859499, 7.08827055361374002488782088399, 7.49268064446268010230997600659, 8.377446654325248117350946284188
