| L(s) = 1 | + 2-s + 5-s + 5·9-s + 10-s − 11-s + 5·18-s + 19-s − 22-s + 29-s − 5·37-s + 38-s − 41-s − 5·43-s + 5·45-s − 47-s + 5·49-s − 53-s − 55-s + 58-s + 61-s − 67-s − 5·74-s + 15·81-s − 82-s − 83-s − 5·86-s + 89-s + ⋯ |
| L(s) = 1 | + 2-s + 5-s + 5·9-s + 10-s − 11-s + 5·18-s + 19-s − 22-s + 29-s − 5·37-s + 38-s − 41-s − 5·43-s + 5·45-s − 47-s + 5·49-s − 53-s − 55-s + 58-s + 61-s − 67-s − 5·74-s + 15·81-s − 82-s − 83-s − 5·86-s + 89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{5} \cdot 43^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{5} \cdot 43^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.636444526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.636444526\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 37 | $C_1$ | \( ( 1 + T )^{5} \) |
| 43 | $C_1$ | \( ( 1 + T )^{5} \) |
| good | 2 | $C_{10}$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 41 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 47 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 53 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 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 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
| 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 | $C_{10}$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{5}( 1 + T )^{5} \) |
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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
−5.62828074279386157134965099066, −5.61206563615957149398494333849, −5.36167273276933951284996825247, −5.25233971080090764621406182212, −5.06389383874739248036040684772, −5.05320243325491222084470633247, −4.66635093012271514565321471692, −4.63811443123084016167131350822, −4.53268929869888821976700990313, −4.45820326517041339953729194224, −3.96499583625845151530390970802, −3.82940574962999756185789908773, −3.66916032220306048369947993582, −3.64765839277683115113315762571, −3.33278496180358473883212519538, −3.29869469710683739027825295686, −2.86051723050027764355951741728, −2.54304098566988989313395239170, −2.26091293328915483550714961267, −2.01257297945463088580094568204, −1.78050209994396952257631775778, −1.64367281085801251695757884868, −1.42235875950857010190497965934, −1.36645364251707590322112703747, −0.853020642368854210908761110985,
0.853020642368854210908761110985, 1.36645364251707590322112703747, 1.42235875950857010190497965934, 1.64367281085801251695757884868, 1.78050209994396952257631775778, 2.01257297945463088580094568204, 2.26091293328915483550714961267, 2.54304098566988989313395239170, 2.86051723050027764355951741728, 3.29869469710683739027825295686, 3.33278496180358473883212519538, 3.64765839277683115113315762571, 3.66916032220306048369947993582, 3.82940574962999756185789908773, 3.96499583625845151530390970802, 4.45820326517041339953729194224, 4.53268929869888821976700990313, 4.63811443123084016167131350822, 4.66635093012271514565321471692, 5.05320243325491222084470633247, 5.06389383874739248036040684772, 5.25233971080090764621406182212, 5.36167273276933951284996825247, 5.61206563615957149398494333849, 5.62828074279386157134965099066
