| L(s) = 1 | − 5-s − 2·7-s + 5·11-s + 4·17-s − 19-s + 25-s − 2·29-s − 7·31-s + 2·35-s + 10·37-s + 41-s + 10·43-s − 6·47-s − 3·49-s − 6·53-s − 5·55-s − 9·61-s + 14·67-s + 5·71-s + 10·73-s − 10·77-s − 11·79-s − 6·83-s − 4·85-s − 10·89-s + 95-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.50·11-s + 0.970·17-s − 0.229·19-s + 1/5·25-s − 0.371·29-s − 1.25·31-s + 0.338·35-s + 1.64·37-s + 0.156·41-s + 1.52·43-s − 0.875·47-s − 3/7·49-s − 0.824·53-s − 0.674·55-s − 1.15·61-s + 1.71·67-s + 0.593·71-s + 1.17·73-s − 1.13·77-s − 1.23·79-s − 0.658·83-s − 0.433·85-s − 1.05·89-s + 0.102·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.502449153\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.502449153\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.41649110778766, −12.23260652053714, −11.42184056925001, −11.16773925742323, −10.92188591304030, −10.01454008961563, −9.712108892492287, −9.354664989705940, −8.989834925321903, −8.364113605403300, −7.821104672848770, −7.511355904238468, −6.873100315564655, −6.482437968899388, −6.088907817102616, −5.555526252101979, −5.004387646515720, −4.288789288776347, −3.836567123636792, −3.636070491501078, −2.888566556376263, −2.453512755853348, −1.467171365864266, −1.209907503975191, −0.3325186187643093,
0.3325186187643093, 1.209907503975191, 1.467171365864266, 2.453512755853348, 2.888566556376263, 3.636070491501078, 3.836567123636792, 4.288789288776347, 5.004387646515720, 5.555526252101979, 6.088907817102616, 6.482437968899388, 6.873100315564655, 7.511355904238468, 7.821104672848770, 8.364113605403300, 8.989834925321903, 9.354664989705940, 9.712108892492287, 10.01454008961563, 10.92188591304030, 11.16773925742323, 11.42184056925001, 12.23260652053714, 12.41649110778766
