| L(s) = 1 | + 3-s + 9-s + 13-s − 19-s + 25-s + 27-s − 31-s − 37-s + 39-s + 43-s − 57-s − 2·61-s + 67-s + 73-s + 75-s + 79-s + 81-s − 93-s − 2·97-s − 103-s − 109-s − 111-s + 117-s + ⋯ |
| L(s) = 1 | + 3-s + 9-s + 13-s − 19-s + 25-s + 27-s − 31-s − 37-s + 39-s + 43-s − 57-s − 2·61-s + 67-s + 73-s + 75-s + 79-s + 81-s − 93-s − 2·97-s − 103-s − 109-s − 111-s + 117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.772975288\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.772975288\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 - T + T^{2} \) |
| 79 | \( 1 - T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 + T )^{2} \) |
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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
−8.973685702351720350172721021791, −8.546235759159742100169669196619, −7.76015759821158699988560862396, −6.93128215755057007946445829916, −6.22202310371934614953862151133, −5.11540985474330752250201617430, −4.13531229289797035801103300267, −3.45841079983394406173556949617, −2.45609020296916569199009014874, −1.41692071422204198446561443685,
1.41692071422204198446561443685, 2.45609020296916569199009014874, 3.45841079983394406173556949617, 4.13531229289797035801103300267, 5.11540985474330752250201617430, 6.22202310371934614953862151133, 6.93128215755057007946445829916, 7.76015759821158699988560862396, 8.546235759159742100169669196619, 8.973685702351720350172721021791
