| L(s) = 1 | + 3-s + 9-s + 4·13-s + 8·19-s − 6·25-s + 27-s + 16·31-s − 12·37-s + 4·39-s − 8·43-s − 14·49-s + 8·57-s + 4·61-s + 8·67-s + 20·73-s − 6·75-s − 16·79-s + 81-s + 16·93-s + 4·97-s + 32·103-s + 4·109-s − 12·111-s + 4·117-s − 6·121-s + 127-s − 8·129-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.10·13-s + 1.83·19-s − 6/5·25-s + 0.192·27-s + 2.87·31-s − 1.97·37-s + 0.640·39-s − 1.21·43-s − 2·49-s + 1.05·57-s + 0.512·61-s + 0.977·67-s + 2.34·73-s − 0.692·75-s − 1.80·79-s + 1/9·81-s + 1.65·93-s + 0.406·97-s + 3.15·103-s + 0.383·109-s − 1.13·111-s + 0.369·117-s − 0.545·121-s + 0.0887·127-s − 0.704·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110592 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110592 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.098868579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.098868579\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−9.780091138821383013974264992430, −8.884556993307409273860919130264, −8.456614023795656741730916500766, −8.157389656224392853374347080198, −7.61947548271568794045481490015, −7.06857822018944724974086155331, −6.38250152569089720234352942221, −6.16826675668887144123811676692, −5.14444399128618341481533912587, −4.99175853408267534988469991920, −4.02227786472289347756607333391, −3.43299575130251299450107440988, −3.04433904291477058087427796742, −1.99999348241168201324552330088, −1.12835632773744922111480113749,
1.12835632773744922111480113749, 1.99999348241168201324552330088, 3.04433904291477058087427796742, 3.43299575130251299450107440988, 4.02227786472289347756607333391, 4.99175853408267534988469991920, 5.14444399128618341481533912587, 6.16826675668887144123811676692, 6.38250152569089720234352942221, 7.06857822018944724974086155331, 7.61947548271568794045481490015, 8.157389656224392853374347080198, 8.456614023795656741730916500766, 8.884556993307409273860919130264, 9.780091138821383013974264992430
