| L(s) = 1 | + 3-s + 5-s + 9-s + 11-s + 13-s + 15-s − 17-s − 19-s + 23-s + 25-s + 27-s − 29-s − 31-s + 33-s + 37-s + 39-s + 41-s + 43-s + 45-s − 51-s + 53-s + 55-s − 57-s + 59-s − 61-s + 65-s + 67-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s + 9-s + 11-s + 13-s + 15-s − 17-s − 19-s + 23-s + 25-s + 27-s − 29-s − 31-s + 33-s + 37-s + 39-s + 41-s + 43-s + 45-s − 51-s + 53-s + 55-s − 57-s + 59-s − 61-s + 65-s + 67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1316 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1316 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.057137843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.057137843\) |
| \(L(1)\) |
\(\approx\) |
\(2.078419315\) |
| \(L(1)\) |
\(\approx\) |
\(2.078419315\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 47 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.769591707217677312816728802153, −20.06975621306696299407019573094, −19.30209822070017878107782864905, −18.52545272663577995936430065668, −17.82493292944367729385484462084, −16.96404621606067986980007281757, −16.214412810174669056433011111966, −15.12109810477518857972485088285, −14.641961775804735763487551635282, −13.83890344231578773104501904879, −13.12042636237981758890243840274, −12.69791808735933800552428672372, −11.205798214247837472480284026252, −10.66519568911979448449920439974, −9.46544156378226501460942458222, −9.08789797895307430532168758031, −8.45088257800492252877736582125, −7.22097306656031854704153188683, −6.52253513991870013652263706948, −5.70658927601644340513302252019, −4.40864764395102956086568345258, −3.76258250438406639017645129197, −2.620005263740170853496235940047, −1.873303302300145181750892288345, −1.00585659589261004015541609543,
1.00585659589261004015541609543, 1.873303302300145181750892288345, 2.620005263740170853496235940047, 3.76258250438406639017645129197, 4.40864764395102956086568345258, 5.70658927601644340513302252019, 6.52253513991870013652263706948, 7.22097306656031854704153188683, 8.45088257800492252877736582125, 9.08789797895307430532168758031, 9.46544156378226501460942458222, 10.66519568911979448449920439974, 11.205798214247837472480284026252, 12.69791808735933800552428672372, 13.12042636237981758890243840274, 13.83890344231578773104501904879, 14.641961775804735763487551635282, 15.12109810477518857972485088285, 16.214412810174669056433011111966, 16.96404621606067986980007281757, 17.82493292944367729385484462084, 18.52545272663577995936430065668, 19.30209822070017878107782864905, 20.06975621306696299407019573094, 20.769591707217677312816728802153
