L(s) = 1 | + 3-s + 6·7-s + 6·11-s − 5·13-s − 3·17-s − 6·19-s + 6·21-s − 6·23-s + 7·25-s − 27-s − 9·29-s + 6·33-s − 9·37-s − 5·39-s − 15·41-s − 2·43-s + 17·49-s − 3·51-s − 18·53-s − 6·57-s + 24·59-s + 11·61-s + 18·67-s − 6·69-s + 18·71-s + 7·75-s + 36·77-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 2.26·7-s + 1.80·11-s − 1.38·13-s − 0.727·17-s − 1.37·19-s + 1.30·21-s − 1.25·23-s + 7/5·25-s − 0.192·27-s − 1.67·29-s + 1.04·33-s − 1.47·37-s − 0.800·39-s − 2.34·41-s − 0.304·43-s + 17/7·49-s − 0.420·51-s − 2.47·53-s − 0.794·57-s + 3.12·59-s + 1.40·61-s + 2.19·67-s − 0.722·69-s + 2.13·71-s + 0.808·75-s + 4.10·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.712976790\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.712976790\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 15 T + 116 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 24 T + 251 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 18 T + 175 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 18 T + 179 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 137 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 12 T + 145 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−12.92969730931887237355674837410, −12.91720984623017817836612683453, −11.86023146607430678276249578473, −11.84788915870168825750305157953, −11.15827558278724835044081022692, −10.96526636825649627149395754262, −9.962720711660750300781971406164, −9.761667962638959940247021542591, −8.711229340482488262986780144509, −8.623581491597419451792796482663, −8.249799728223798369499347931976, −7.44773090777286749523056087306, −6.86948085074299929749018448082, −6.45984664369464484242152134746, −5.16726744818237835871522606195, −5.02783433508979592011012781687, −4.16420577549207164277272409888, −3.64827924370899148887378178731, −2.04421677189285670853746386208, −1.87434699414572112069597478785,
1.87434699414572112069597478785, 2.04421677189285670853746386208, 3.64827924370899148887378178731, 4.16420577549207164277272409888, 5.02783433508979592011012781687, 5.16726744818237835871522606195, 6.45984664369464484242152134746, 6.86948085074299929749018448082, 7.44773090777286749523056087306, 8.249799728223798369499347931976, 8.623581491597419451792796482663, 8.711229340482488262986780144509, 9.761667962638959940247021542591, 9.962720711660750300781971406164, 10.96526636825649627149395754262, 11.15827558278724835044081022692, 11.84788915870168825750305157953, 11.86023146607430678276249578473, 12.91720984623017817836612683453, 12.92969730931887237355674837410