L(s) = 1 | + 3·3-s + 3·5-s + 4·9-s + 2·11-s + 10·13-s + 9·15-s + 2·17-s + 3·19-s − 3·23-s + 6·27-s + 17·29-s − 11·31-s + 6·33-s + 6·37-s + 30·39-s + 2·41-s − 16·43-s + 12·45-s + 18·47-s + 6·51-s − 9·53-s + 6·55-s + 9·57-s − 59-s + 6·61-s + 30·65-s − 7·67-s + ⋯ |
L(s) = 1 | + 1.73·3-s + 1.34·5-s + 4/3·9-s + 0.603·11-s + 2.77·13-s + 2.32·15-s + 0.485·17-s + 0.688·19-s − 0.625·23-s + 1.15·27-s + 3.15·29-s − 1.97·31-s + 1.04·33-s + 0.986·37-s + 4.80·39-s + 0.312·41-s − 2.43·43-s + 1.78·45-s + 2.62·47-s + 0.840·51-s − 1.23·53-s + 0.809·55-s + 1.19·57-s − 0.130·59-s + 0.768·61-s + 3.72·65-s − 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64577296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64577296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(13.65970371\) |
\(L(\frac12)\) |
\(\approx\) |
\(13.65970371\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $C_4$ | \( 1 - p T + 5 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 3 T + 9 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 2 T - p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 11 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 45 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 17 T + 127 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 11 T + 63 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 16 T + 137 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 123 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 89 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 79 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 117 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 10 T + 115 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 26 T + 314 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 97 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 7 T + 47 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−8.188321110831117079718253589436, −7.71638990036421097168800748589, −7.36320937280697382074233550496, −7.09821661694762481092699688293, −6.46972429020839511798112126250, −6.20371412498216636535884655763, −6.06424743798278492719366442306, −5.74120433421869074987341278352, −5.31737732229422706927559206562, −4.75066671987062043245986007123, −4.18394956186334251089442632284, −4.16267759114975335419251961844, −3.38708661925172239682050548395, −3.34228521515358622384660519204, −2.96071605181874240499387731422, −2.53635105582756006745090762891, −1.96087733388379321563108560913, −1.55127218323864400537635803381, −1.31086234224679349344492469723, −0.78447760048948505192138535218,
0.78447760048948505192138535218, 1.31086234224679349344492469723, 1.55127218323864400537635803381, 1.96087733388379321563108560913, 2.53635105582756006745090762891, 2.96071605181874240499387731422, 3.34228521515358622384660519204, 3.38708661925172239682050548395, 4.16267759114975335419251961844, 4.18394956186334251089442632284, 4.75066671987062043245986007123, 5.31737732229422706927559206562, 5.74120433421869074987341278352, 6.06424743798278492719366442306, 6.20371412498216636535884655763, 6.46972429020839511798112126250, 7.09821661694762481092699688293, 7.36320937280697382074233550496, 7.71638990036421097168800748589, 8.188321110831117079718253589436