| L(s) = 1 | + 4·7-s − 2·19-s + 8·25-s − 20·29-s + 20·41-s + 24·43-s − 2·49-s + 20·53-s − 24·59-s + 16·61-s − 16·71-s + 12·73-s + 12·89-s − 16·107-s + 12·113-s + 4·121-s + 127-s + 131-s − 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 18·169-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.458·19-s + 8/5·25-s − 3.71·29-s + 3.12·41-s + 3.65·43-s − 2/7·49-s + 2.74·53-s − 3.12·59-s + 2.04·61-s − 1.89·71-s + 1.40·73-s + 1.27·89-s − 1.54·107-s + 1.12·113-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s − 0.693·133-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 + 1.38·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1871424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.805647288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.805647288\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 92 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 148 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 122 T^{2} + p^{2} T^{4} \) |
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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
−9.442187623633944833672928784863, −9.402570657259317195297189230545, −9.028401097517210982921788062384, −8.700336458057566366144449959824, −8.075771189874845261506942247365, −7.68440395351289788178872173461, −7.36055389012067583927051587305, −7.31226398822311594873483251285, −6.45817429480847979334115785106, −5.94722235434439467794277978307, −5.53403863207904805216335855968, −5.38833416481285594707329183524, −4.56176659526076967099782153703, −4.30487931468662572123154714922, −3.94977891689844462860883213180, −3.26081149359672097723831814931, −2.37413099458576657242682178528, −2.25532421018789825959488073162, −1.40742200417023422299684696570, −0.73690625139353493193060173269,
0.73690625139353493193060173269, 1.40742200417023422299684696570, 2.25532421018789825959488073162, 2.37413099458576657242682178528, 3.26081149359672097723831814931, 3.94977891689844462860883213180, 4.30487931468662572123154714922, 4.56176659526076967099782153703, 5.38833416481285594707329183524, 5.53403863207904805216335855968, 5.94722235434439467794277978307, 6.45817429480847979334115785106, 7.31226398822311594873483251285, 7.36055389012067583927051587305, 7.68440395351289788178872173461, 8.075771189874845261506942247365, 8.700336458057566366144449959824, 9.028401097517210982921788062384, 9.402570657259317195297189230545, 9.442187623633944833672928784863
