L(s) = 1 | + 2·9-s − 8·11-s − 2·19-s + 12·29-s − 16·31-s − 4·41-s + 14·49-s + 16·59-s + 4·61-s − 16·71-s + 8·79-s − 5·81-s + 36·89-s − 16·99-s + 20·101-s + 4·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯ |
L(s) = 1 | + 2/3·9-s − 2.41·11-s − 0.458·19-s + 2.22·29-s − 2.87·31-s − 0.624·41-s + 2·49-s + 2.08·59-s + 0.512·61-s − 1.89·71-s + 0.900·79-s − 5/9·81-s + 3.81·89-s − 1.60·99-s + 1.99·101-s + 0.383·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14440000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.564318055\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.564318055\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−8.600490612133485140072493170118, −8.368594485502377344560929485343, −7.79429008931522625111058771492, −7.67473378577160993432617844906, −7.19083435255323682069450919347, −7.03026440298664264605205141397, −6.53129176987406667258668231547, −5.92802227178778727402680733423, −5.74021814745134973998248861979, −5.25087172880662677657622445678, −4.81891976734193592567706230130, −4.79337381446210053928085388874, −3.99008239393249547241255919867, −3.72339528297593534327338333426, −3.15642961271282818174487236214, −2.70418186520687253034719450080, −2.09514546165216426893649880899, −2.06889194092672978034692285573, −1.02663657397353437532918763041, −0.40693001706152089466805959728,
0.40693001706152089466805959728, 1.02663657397353437532918763041, 2.06889194092672978034692285573, 2.09514546165216426893649880899, 2.70418186520687253034719450080, 3.15642961271282818174487236214, 3.72339528297593534327338333426, 3.99008239393249547241255919867, 4.79337381446210053928085388874, 4.81891976734193592567706230130, 5.25087172880662677657622445678, 5.74021814745134973998248861979, 5.92802227178778727402680733423, 6.53129176987406667258668231547, 7.03026440298664264605205141397, 7.19083435255323682069450919347, 7.67473378577160993432617844906, 7.79429008931522625111058771492, 8.368594485502377344560929485343, 8.600490612133485140072493170118