| L(s) = 1 | + 5·2-s − 3-s + 15·4-s − 5-s − 5·6-s + 35·8-s − 5·10-s − 11-s − 15·12-s − 13-s + 15-s + 70·16-s − 19-s − 15·20-s − 5·22-s − 35·24-s − 5·26-s − 29-s + 5·30-s + 126·32-s + 33-s − 37-s − 5·38-s + 39-s − 35·40-s − 41-s − 15·44-s + ⋯ |
| L(s) = 1 | + 5·2-s − 3-s + 15·4-s − 5-s − 5·6-s + 35·8-s − 5·10-s − 11-s − 15·12-s − 13-s + 15-s + 70·16-s − 19-s − 15·20-s − 5·22-s − 35·24-s − 5·26-s − 29-s + 5·30-s + 126·32-s + 33-s − 37-s − 5·38-s + 39-s − 35·40-s − 41-s − 15·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 163^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 163^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(17.99354384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.99354384\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{5} \) |
| 163 | $C_1$ | \( ( 1 - T )^{5} \) |
| good | 3 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 5 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 11 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 13 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 19 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 29 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 37 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 41 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 47 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 59 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 67 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 71 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 97 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.69550458013521488575610389028, −5.67339253879251392499874973422, −5.58711839830927816182622788364, −5.45369651440744799427916734818, −5.26986216128255403070173974495, −5.16205928780752946834742037975, −5.12511175921664206642588528737, −4.59163045550744044033544825358, −4.54744526291529868957369200104, −4.38678492240882684908836490886, −4.24103529500060714738287590218, −4.04191907951308337052558724396, −3.94200282677566409564013445744, −3.78860019094056842198598593270, −3.43149195583217359391445643443, −3.22506122010674913647964370861, −3.12116726266440893801343968522, −2.76287056834018055160439405415, −2.69534322685201740588308474433, −2.55433929013862000853122646330, −2.05378594073999999068355458779, −1.97478062792938487520406885512, −1.93455060978440383216460627740, −1.34508569576849823377576793347, −1.11302435206381325298402161645,
1.11302435206381325298402161645, 1.34508569576849823377576793347, 1.93455060978440383216460627740, 1.97478062792938487520406885512, 2.05378594073999999068355458779, 2.55433929013862000853122646330, 2.69534322685201740588308474433, 2.76287056834018055160439405415, 3.12116726266440893801343968522, 3.22506122010674913647964370861, 3.43149195583217359391445643443, 3.78860019094056842198598593270, 3.94200282677566409564013445744, 4.04191907951308337052558724396, 4.24103529500060714738287590218, 4.38678492240882684908836490886, 4.54744526291529868957369200104, 4.59163045550744044033544825358, 5.12511175921664206642588528737, 5.16205928780752946834742037975, 5.26986216128255403070173974495, 5.45369651440744799427916734818, 5.58711839830927816182622788364, 5.67339253879251392499874973422, 5.69550458013521488575610389028
