L(s) = 1 | − 7-s − 4·9-s + 4·11-s + 4·23-s + 8·25-s + 6·29-s + 2·37-s + 4·43-s + 49-s + 6·53-s + 4·63-s + 16·67-s − 4·77-s + 7·81-s − 16·99-s − 14·109-s + 8·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4·161-s + 163-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 4/3·9-s + 1.20·11-s + 0.834·23-s + 8/5·25-s + 1.11·29-s + 0.328·37-s + 0.609·43-s + 1/7·49-s + 0.824·53-s + 0.503·63-s + 1.95·67-s − 0.455·77-s + 7/9·81-s − 1.60·99-s − 1.34·109-s + 0.752·113-s − 0.545·121-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 + 0.0798·157-s − 0.315·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.066629834\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.066629834\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
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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
−8.070554277958608681067069901951, −7.45534480510550780600894978682, −6.87809379436962876194139034037, −6.70596170375642853522656426400, −6.30133104588686386649234355769, −5.72536085675769033661150029796, −5.40514267925738802246364291195, −4.78259191210460700938369816010, −4.43229698035645280760312798128, −3.68790252762927401658763250807, −3.34687522568984549173258021610, −2.70264903304100365711098848140, −2.39816746526115311876061747224, −1.28223936846994640878865622697, −0.69410686200119619100891679442,
0.69410686200119619100891679442, 1.28223936846994640878865622697, 2.39816746526115311876061747224, 2.70264903304100365711098848140, 3.34687522568984549173258021610, 3.68790252762927401658763250807, 4.43229698035645280760312798128, 4.78259191210460700938369816010, 5.40514267925738802246364291195, 5.72536085675769033661150029796, 6.30133104588686386649234355769, 6.70596170375642853522656426400, 6.87809379436962876194139034037, 7.45534480510550780600894978682, 8.070554277958608681067069901951