| L(s) = 1 | − 2-s + 5·3-s − 5-s − 5·6-s − 7-s + 15·9-s + 10-s − 11-s + 14-s − 5·15-s − 17-s − 15·18-s − 19-s − 5·21-s + 22-s + 35·27-s + 5·30-s − 5·33-s + 34-s + 35-s − 37-s + 38-s + 5·42-s − 43-s − 15·45-s − 5·51-s − 35·54-s + ⋯ |
| L(s) = 1 | − 2-s + 5·3-s − 5-s − 5·6-s − 7-s + 15·9-s + 10-s − 11-s + 14-s − 5·15-s − 17-s − 15·18-s − 19-s − 5·21-s + 22-s + 35·27-s + 5·30-s − 5·33-s + 34-s + 35-s − 37-s + 38-s + 5·42-s − 43-s − 15·45-s − 5·51-s − 35·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 197^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{5} \cdot 197^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.240864531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.240864531\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{5} \) |
| 197 | $C_1$ | \( ( 1 - T )^{5} \) |
| good | 2 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 5 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 7 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 17 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 43 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 61 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 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
−6.90982890423019141803039018365, −6.87179964416648458582562903539, −6.83835463497953047176918864658, −6.32919895151901999340199458956, −6.31538538301541073400568703538, −5.99257724283257570232639670503, −5.42369569191795396236866110225, −5.27126801929357672670714811071, −4.89246384572632318309812830885, −4.88055421624727734066880052540, −4.40036972344417240334454856703, −4.18550701066793006011325118947, −4.16243177774988857354199660172, −4.15553796664477471072882470712, −3.61001271427431092943992928193, −3.46464620374451226359127263808, −3.37579794647086645013551405443, −3.08302653378897140921481583928, −2.87167538293516623059451299215, −2.61041148005977183526182438709, −2.35137057756022469972169375500, −2.31264171519099436316359024194, −1.82708601027781497052058655860, −1.49978973244071203653125346655, −1.33724585627002362314425100504,
1.33724585627002362314425100504, 1.49978973244071203653125346655, 1.82708601027781497052058655860, 2.31264171519099436316359024194, 2.35137057756022469972169375500, 2.61041148005977183526182438709, 2.87167538293516623059451299215, 3.08302653378897140921481583928, 3.37579794647086645013551405443, 3.46464620374451226359127263808, 3.61001271427431092943992928193, 4.15553796664477471072882470712, 4.16243177774988857354199660172, 4.18550701066793006011325118947, 4.40036972344417240334454856703, 4.88055421624727734066880052540, 4.89246384572632318309812830885, 5.27126801929357672670714811071, 5.42369569191795396236866110225, 5.99257724283257570232639670503, 6.31538538301541073400568703538, 6.32919895151901999340199458956, 6.83835463497953047176918864658, 6.87179964416648458582562903539, 6.90982890423019141803039018365
