L(s) = 1 | − 9-s − 8·11-s − 2·19-s − 6·31-s − 12·41-s + 5·49-s − 12·59-s + 2·61-s + 12·71-s + 16·79-s + 81-s + 32·89-s + 8·99-s + 4·101-s + 34·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 23·169-s + 2·171-s + ⋯ |
L(s) = 1 | − 1/3·9-s − 2.41·11-s − 0.458·19-s − 1.07·31-s − 1.87·41-s + 5/7·49-s − 1.56·59-s + 0.256·61-s + 1.42·71-s + 1.80·79-s + 1/9·81-s + 3.39·89-s + 0.804·99-s + 0.398·101-s + 3.25·109-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 − 1.76·169-s + 0.152·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23040000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4557082633\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4557082633\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
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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
−8.380260812241747118382121829106, −8.159779697140662572885214661941, −7.68598355571059259400808525658, −7.42342037867943676798386343712, −7.23886851908443524052817575742, −6.60208643870057916332484204761, −6.17804512234843785956903368192, −6.01572360571272414893579352776, −5.43853808257547654783323041771, −5.00747593620171341378783529598, −4.98398249346912774640873582323, −4.59838225143650363635330529575, −3.77173174418220382482328929622, −3.46461777269975231687720824863, −3.22858301727305229441128536217, −2.37708387311314904406939940019, −2.36997253810835934028763502311, −1.86838946400458614185275649069, −0.991589173609226191602957495501, −0.20106195655067416479827283333,
0.20106195655067416479827283333, 0.991589173609226191602957495501, 1.86838946400458614185275649069, 2.36997253810835934028763502311, 2.37708387311314904406939940019, 3.22858301727305229441128536217, 3.46461777269975231687720824863, 3.77173174418220382482328929622, 4.59838225143650363635330529575, 4.98398249346912774640873582323, 5.00747593620171341378783529598, 5.43853808257547654783323041771, 6.01572360571272414893579352776, 6.17804512234843785956903368192, 6.60208643870057916332484204761, 7.23886851908443524052817575742, 7.42342037867943676798386343712, 7.68598355571059259400808525658, 8.159779697140662572885214661941, 8.380260812241747118382121829106