L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 5-s + 2·6-s − 4·7-s − 2·9-s − 2·10-s + 2·11-s − 2·12-s + 8·14-s − 15-s − 4·16-s − 4·17-s + 4·18-s + 2·20-s + 4·21-s − 4·22-s − 4·25-s + 5·27-s − 8·28-s + 2·30-s + 8·32-s − 2·33-s + 8·34-s − 4·35-s − 4·36-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 1.51·7-s − 2/3·9-s − 0.632·10-s + 0.603·11-s − 0.577·12-s + 2.13·14-s − 0.258·15-s − 16-s − 0.970·17-s + 0.942·18-s + 0.447·20-s + 0.872·21-s − 0.852·22-s − 4/5·25-s + 0.962·27-s − 1.51·28-s + 0.365·30-s + 1.41·32-s − 0.348·33-s + 1.37·34-s − 0.676·35-s − 2/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2079652962\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2079652962\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.603539619290756001226038948684, −8.399321207327632052846971537333, −7.79929959554017504493480383489, −7.11798617010190808321460327259, −6.75700481068273204827606843552, −6.36261389471308870138602900888, −6.16530520176828730622789022769, −5.41325790295005799624258361008, −4.92404269467477878351185887984, −4.19846804805603804873348477595, −3.55134596504967876675656965993, −2.91823833046493033163603826267, −2.22901314573716685967671256577, −1.47396545601602715073166640408, −0.32265848357517983832832095703,
0.32265848357517983832832095703, 1.47396545601602715073166640408, 2.22901314573716685967671256577, 2.91823833046493033163603826267, 3.55134596504967876675656965993, 4.19846804805603804873348477595, 4.92404269467477878351185887984, 5.41325790295005799624258361008, 6.16530520176828730622789022769, 6.36261389471308870138602900888, 6.75700481068273204827606843552, 7.11798617010190808321460327259, 7.79929959554017504493480383489, 8.399321207327632052846971537333, 8.603539619290756001226038948684