| L(s) = 1 | + 7-s − 3·13-s − 25-s + 2·37-s + 2·43-s − 3·61-s + 67-s + 79-s − 3·91-s + 3·97-s + 3·103-s − 4·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 173-s − 175-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 7-s − 3·13-s − 25-s + 2·37-s + 2·43-s − 3·61-s + 67-s + 79-s − 3·91-s + 3·97-s + 3·103-s − 4·109-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 5·169-s + 173-s − 175-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.096798734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.096798734\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.420056535421049446472641110578, −9.165248785724656983582651011530, −8.642851317633414069579636949229, −8.006807662205299319200684831470, −7.78153581652768383801341344239, −7.61235372126608453422466007365, −7.23799822220650440798619854488, −6.85433948495754649695117689604, −6.12118354656560318744628813082, −5.90758592807487326334319707482, −5.46934323291334221500532674465, −4.75202940456689191078574519267, −4.64567970474956593744742090617, −4.51977472756465596439625184697, −3.72391015436283279827454019357, −3.12270504489969288711812826615, −2.47442362313862663710996778273, −2.28481665526116421541602402999, −1.73068103862767334495426726424, −0.71197459327781752028770392355,
0.71197459327781752028770392355, 1.73068103862767334495426726424, 2.28481665526116421541602402999, 2.47442362313862663710996778273, 3.12270504489969288711812826615, 3.72391015436283279827454019357, 4.51977472756465596439625184697, 4.64567970474956593744742090617, 4.75202940456689191078574519267, 5.46934323291334221500532674465, 5.90758592807487326334319707482, 6.12118354656560318744628813082, 6.85433948495754649695117689604, 7.23799822220650440798619854488, 7.61235372126608453422466007365, 7.78153581652768383801341344239, 8.006807662205299319200684831470, 8.642851317633414069579636949229, 9.165248785724656983582651011530, 9.420056535421049446472641110578
