| L(s) = 1 | − 9-s − 12·11-s + 10·19-s − 12·29-s − 2·31-s + 13·49-s − 12·59-s + 26·61-s − 16·79-s + 81-s + 12·99-s + 24·101-s − 14·109-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 10·171-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 3.61·11-s + 2.29·19-s − 2.22·29-s − 0.359·31-s + 13/7·49-s − 1.56·59-s + 3.32·61-s − 1.80·79-s + 1/9·81-s + 1.20·99-s + 2.38·101-s − 1.34·109-s + 7.81·121-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 + 1/13·169-s − 0.764·171-s + 0.0760·173-s + 0.0747·179-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
| \(L(1)\) |
\(\approx\) |
\(0.8809552443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8809552443\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−8.263036196814095423426182857056, −8.020302983398762490374308801290, −7.74423522548539021008573965336, −7.42647442619073308913700446337, −7.16991709322426984358270463569, −6.94543886427299249698296477162, −6.01942560838314896012110546510, −5.65814781137762651310429143482, −5.59937913490160382905881450944, −5.23800960721941197514360612449, −4.98147381913464445755744155115, −4.52224322818064278823693697505, −3.74455416746621122666202831079, −3.57592167631857452520778280353, −2.83558226706090182571971106320, −2.83090353838038663148841739265, −2.25317848805833607774751430387, −1.83893467809733139436190253683, −0.935542377311310171069205104773, −0.29430172760439236458916789589,
0.29430172760439236458916789589, 0.935542377311310171069205104773, 1.83893467809733139436190253683, 2.25317848805833607774751430387, 2.83090353838038663148841739265, 2.83558226706090182571971106320, 3.57592167631857452520778280353, 3.74455416746621122666202831079, 4.52224322818064278823693697505, 4.98147381913464445755744155115, 5.23800960721941197514360612449, 5.59937913490160382905881450944, 5.65814781137762651310429143482, 6.01942560838314896012110546510, 6.94543886427299249698296477162, 7.16991709322426984358270463569, 7.42647442619073308913700446337, 7.74423522548539021008573965336, 8.020302983398762490374308801290, 8.263036196814095423426182857056
