| L(s) = 1 | + 4·7-s + 6·19-s + 5·25-s − 3·31-s − 37-s + 26·43-s + 9·49-s + 27·61-s − 11·67-s − 24·73-s + 13·79-s − 33·103-s − 19·109-s − 11·121-s + 127-s + 131-s + 24·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 23·169-s + 173-s + 20·175-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.37·19-s + 25-s − 0.538·31-s − 0.164·37-s + 3.96·43-s + 9/7·49-s + 3.45·61-s − 1.34·67-s − 2.80·73-s + 1.46·79-s − 3.25·103-s − 1.81·109-s − 121-s + 0.0887·127-s + 0.0873·131-s + 2.08·133-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 + 1.76·169-s + 0.0760·173-s + 1.51·175-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.691417628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.691417628\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.53221965134939149904171381490, −10.37059750720125497271804065196, −9.427848159944917642142647213695, −9.419036510147655251564257092153, −8.825278386768614158005947042004, −8.444822191514103534905182382007, −7.86676582017030471190609342934, −7.66345269815501757713572860055, −7.06981337675508768949455224979, −6.89937837502719350405088009205, −5.87776417435448985938693139540, −5.71067950860799550435907496887, −5.13269741258244896107698922896, −4.79236386620650775355487539075, −4.05741181497139106535100787429, −3.83043812705157068655464957497, −2.74675457327283956911640880157, −2.50729415977030945124102090545, −1.47871982250095917530485152702, −0.959911025670626443655042082832,
0.959911025670626443655042082832, 1.47871982250095917530485152702, 2.50729415977030945124102090545, 2.74675457327283956911640880157, 3.83043812705157068655464957497, 4.05741181497139106535100787429, 4.79236386620650775355487539075, 5.13269741258244896107698922896, 5.71067950860799550435907496887, 5.87776417435448985938693139540, 6.89937837502719350405088009205, 7.06981337675508768949455224979, 7.66345269815501757713572860055, 7.86676582017030471190609342934, 8.444822191514103534905182382007, 8.825278386768614158005947042004, 9.419036510147655251564257092153, 9.427848159944917642142647213695, 10.37059750720125497271804065196, 10.53221965134939149904171381490
