| L(s) = 1 | − 2-s + 2·5-s + 8-s + 9-s − 2·10-s − 13-s − 16-s − 18-s − 19-s − 23-s + 3·25-s + 26-s − 37-s + 38-s + 2·40-s + 2·43-s + 2·45-s + 46-s − 47-s + 49-s − 3·50-s − 53-s − 59-s − 61-s + 64-s − 2·65-s + 2·71-s + ⋯ |
| L(s) = 1 | − 2-s + 2·5-s + 8-s + 9-s − 2·10-s − 13-s − 16-s − 18-s − 19-s − 23-s + 3·25-s + 26-s − 37-s + 38-s + 2·40-s + 2·43-s + 2·45-s + 46-s − 47-s + 49-s − 3·50-s − 53-s − 59-s − 61-s + 64-s − 2·65-s + 2·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7630824096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7630824096\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 751 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )^{2} \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.37411297384086791138984206121, −9.523991718450490963596057285827, −9.299543598030750723173832160280, −8.159808289538594989031650160088, −7.13496330291338324396068118187, −6.32381310236204851262967904080, −5.24299974392866300435388282116, −4.34440863020886693130280326015, −2.35311140940197191196905622498, −1.54292585928936861961302471346,
1.54292585928936861961302471346, 2.35311140940197191196905622498, 4.34440863020886693130280326015, 5.24299974392866300435388282116, 6.32381310236204851262967904080, 7.13496330291338324396068118187, 8.159808289538594989031650160088, 9.299543598030750723173832160280, 9.523991718450490963596057285827, 10.37411297384086791138984206121
