| L(s) = 1 | − 2·7-s − 4·13-s + 4·19-s + 4·31-s + 2·37-s − 10·43-s + 3·49-s + 16·61-s + 14·67-s − 16·73-s + 10·79-s + 8·91-s + 20·97-s − 4·103-s − 2·109-s + 5·121-s + 127-s + 131-s − 8·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.10·13-s + 0.917·19-s + 0.718·31-s + 0.328·37-s − 1.52·43-s + 3/7·49-s + 2.04·61-s + 1.71·67-s − 1.87·73-s + 1.12·79-s + 0.838·91-s + 2.03·97-s − 0.394·103-s − 0.191·109-s + 5/11·121-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 − 1.07·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39690000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.524579939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.524579939\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.079827647138800709655667568088, −8.009453558555650146854530908810, −7.29052118447827194730304903333, −7.24864177144473226148728766918, −6.83274747774986761647498516345, −6.53653295086721472746579570588, −6.11635716448353117977695011511, −5.73657458636930658568946142645, −5.23405696528098221058433277587, −5.15933903721764150552906732447, −4.56936313411491332175208358602, −4.31082379899903763448916238423, −3.70105857299602315751269251191, −3.43519736911905592307639602491, −2.82698501253899719782525062103, −2.80070195812088469099053808016, −1.96244211369667752975547251027, −1.80252283893987559868457194371, −0.75048225872808911157737227977, −0.56108504410263755734081397261,
0.56108504410263755734081397261, 0.75048225872808911157737227977, 1.80252283893987559868457194371, 1.96244211369667752975547251027, 2.80070195812088469099053808016, 2.82698501253899719782525062103, 3.43519736911905592307639602491, 3.70105857299602315751269251191, 4.31082379899903763448916238423, 4.56936313411491332175208358602, 5.15933903721764150552906732447, 5.23405696528098221058433277587, 5.73657458636930658568946142645, 6.11635716448353117977695011511, 6.53653295086721472746579570588, 6.83274747774986761647498516345, 7.24864177144473226148728766918, 7.29052118447827194730304903333, 8.009453558555650146854530908810, 8.079827647138800709655667568088
