L(s) = 1 | − 4-s − 9-s + 4·11-s + 16-s − 8·19-s − 16·31-s + 36-s − 4·41-s − 4·44-s + 14·49-s − 4·61-s − 64-s − 20·71-s + 8·76-s + 81-s − 8·89-s − 4·99-s − 4·109-s − 10·121-s + 16·124-s + 127-s + 131-s + 137-s + 139-s − 144-s + 149-s + 151-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 1/3·9-s + 1.20·11-s + 1/4·16-s − 1.83·19-s − 2.87·31-s + 1/6·36-s − 0.624·41-s − 0.603·44-s + 2·49-s − 0.512·61-s − 1/8·64-s − 2.37·71-s + 0.917·76-s + 1/9·81-s − 0.847·89-s − 0.402·99-s − 0.383·109-s − 0.909·121-s + 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.0833·144-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5277375936\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5277375936\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 23 | $C_2$ | \( 1 + T^{2} \) |
good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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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
−8.800841106865568606248544608588, −8.671708303761309882348714750303, −8.078056624496485883520350704103, −7.57687396507234789301980145130, −7.40359581359802599289776427613, −6.82413785473615188906291682597, −6.63068410472462867595353760249, −6.07897520738141816143513653911, −5.83156591356276053829149837625, −5.38805820420280999529783973466, −5.03493033654997584626407117887, −4.31560799559786495352568316536, −4.24475847355889004858784986094, −3.67155124745838427436242287520, −3.53292860288006180194309143443, −2.69948330882332794981554872726, −2.30796909921570149753098045218, −1.64447752652188570346616505164, −1.30215581999366280544888684773, −0.22235180309532984340696959816,
0.22235180309532984340696959816, 1.30215581999366280544888684773, 1.64447752652188570346616505164, 2.30796909921570149753098045218, 2.69948330882332794981554872726, 3.53292860288006180194309143443, 3.67155124745838427436242287520, 4.24475847355889004858784986094, 4.31560799559786495352568316536, 5.03493033654997584626407117887, 5.38805820420280999529783973466, 5.83156591356276053829149837625, 6.07897520738141816143513653911, 6.63068410472462867595353760249, 6.82413785473615188906291682597, 7.40359581359802599289776427613, 7.57687396507234789301980145130, 8.078056624496485883520350704103, 8.671708303761309882348714750303, 8.800841106865568606248544608588