This L-function has the smallest analytic conductor among primitive algebraic degree 2 L-functions.
| L(s) = 1 | − 2-s − 3-s + 6-s + 8-s − 13-s − 16-s + 23-s − 24-s + 25-s + 26-s + 27-s − 29-s − 31-s + 39-s − 41-s − 46-s − 47-s + 48-s + 49-s − 50-s − 54-s + 58-s + 2·59-s + 62-s + 64-s − 69-s − 71-s + ⋯ |
| L(s) = 1 | − 2-s − 3-s + 6-s + 8-s − 13-s − 16-s + 23-s − 24-s + 25-s + 26-s + 27-s − 29-s − 31-s + 39-s − 41-s − 46-s − 47-s + 48-s + 49-s − 50-s − 54-s + 58-s + 2·59-s + 62-s + 64-s − 69-s − 71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1740363269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1740363269\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( 1 + T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−18.08066174344421407705280597229, −17.10126330358183155194872344463, −16.49823251336240134157531665336, −14.66248095457465912598048536284, −12.93440966771841183579118145714, −11.43003635304813311896205990224, −10.28202740085213205371618003800, −8.881396573689177474965824491608, −7.15926229054170384129282782025, −5.11568332881511759855335642038,
5.11568332881511759855335642038, 7.15926229054170384129282782025, 8.881396573689177474965824491608, 10.28202740085213205371618003800, 11.43003635304813311896205990224, 12.93440966771841183579118145714, 14.66248095457465912598048536284, 16.49823251336240134157531665336, 17.10126330358183155194872344463, 18.08066174344421407705280597229
