| L(s) = 1 | − 3·9-s − 2·11-s − 8·19-s + 16·29-s − 14·31-s + 8·41-s + 10·49-s + 2·59-s + 8·61-s + 6·71-s − 4·79-s − 30·89-s + 6·99-s − 20·101-s + 28·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 24·171-s + ⋯ |
| L(s) = 1 | − 9-s − 0.603·11-s − 1.83·19-s + 2.97·29-s − 2.51·31-s + 1.24·41-s + 10/7·49-s + 0.260·59-s + 1.02·61-s + 0.712·71-s − 0.450·79-s − 3.17·89-s + 0.603·99-s − 1.99·101-s + 2.68·109-s + 3/11·121-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 + 2·169-s + 1.83·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.096170988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.096170988\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−9.100300216802742681981262946203, −8.607736628451271902192117582668, −8.521906515164382943475503440682, −8.405694847337583240332319627412, −7.56383299680926231065162165924, −7.49184241046276286777154528319, −6.75829813671257692859733827703, −6.63468824755675029160922967536, −6.08008925938768878080423438816, −5.55537089147830317549722857115, −5.51155788214468242046076751569, −4.85929283512734445722717393183, −4.23647158915717059994512065236, −4.17553723749593452804927274421, −3.42080114523957512133246048914, −2.89315977376610542255111750131, −2.43578108182827528965647095380, −2.14580447153418266999785570821, −1.22377971679948036065245449994, −0.37836928282661036551724975602,
0.37836928282661036551724975602, 1.22377971679948036065245449994, 2.14580447153418266999785570821, 2.43578108182827528965647095380, 2.89315977376610542255111750131, 3.42080114523957512133246048914, 4.17553723749593452804927274421, 4.23647158915717059994512065236, 4.85929283512734445722717393183, 5.51155788214468242046076751569, 5.55537089147830317549722857115, 6.08008925938768878080423438816, 6.63468824755675029160922967536, 6.75829813671257692859733827703, 7.49184241046276286777154528319, 7.56383299680926231065162165924, 8.405694847337583240332319627412, 8.521906515164382943475503440682, 8.607736628451271902192117582668, 9.100300216802742681981262946203
