| L(s) = 1 | + 2·7-s − 2·17-s − 10·23-s + 6·25-s + 14·31-s − 4·41-s + 3·49-s − 30·71-s + 28·79-s + 26·89-s − 28·97-s + 6·103-s − 12·113-s − 4·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 20·161-s + 163-s + 167-s + 169-s + 173-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.485·17-s − 2.08·23-s + 6/5·25-s + 2.51·31-s − 0.624·41-s + 3/7·49-s − 3.56·71-s + 3.15·79-s + 2.75·89-s − 2.84·97-s + 0.591·103-s − 1.12·113-s − 0.366·119-s + 2·121-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 − 1.57·161-s + 0.0783·163-s + 0.0773·167-s + 1/13·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36578304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.865241217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.865241217\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.377838057258067812526962422941, −8.052005362050497021357290274740, −7.41189322615065550503117546900, −7.39351405520343852052235233563, −6.80673350326267440212694135962, −6.39649064168700851618727296672, −6.04814665091131104982570774359, −6.04250512282275106407261646131, −5.15615997851308991896631190871, −5.08990941333837346292580766516, −4.60778178362624913894369933707, −4.28808132185903897877814319405, −3.97668322015864853930066019495, −3.44258532200000085710139868701, −2.80647286765662166920991415580, −2.71974295480374889164080691186, −1.85179826352657479286814848016, −1.84123459574064292768280275779, −1.00138193297008511099446988485, −0.48488818372448211686100751049,
0.48488818372448211686100751049, 1.00138193297008511099446988485, 1.84123459574064292768280275779, 1.85179826352657479286814848016, 2.71974295480374889164080691186, 2.80647286765662166920991415580, 3.44258532200000085710139868701, 3.97668322015864853930066019495, 4.28808132185903897877814319405, 4.60778178362624913894369933707, 5.08990941333837346292580766516, 5.15615997851308991896631190871, 6.04250512282275106407261646131, 6.04814665091131104982570774359, 6.39649064168700851618727296672, 6.80673350326267440212694135962, 7.39351405520343852052235233563, 7.41189322615065550503117546900, 8.052005362050497021357290274740, 8.377838057258067812526962422941
