| L(s) = 1 | + 2-s − 3-s + 4-s − 3.17·5-s − 6-s − 4.05·7-s + 8-s + 9-s − 3.17·10-s + 2.38·11-s − 12-s + 0.293·13-s − 4.05·14-s + 3.17·15-s + 16-s − 4.81·17-s + 18-s − 4.05·19-s − 3.17·20-s + 4.05·21-s + 2.38·22-s + 23-s − 24-s + 5.05·25-s + 0.293·26-s − 27-s − 4.05·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.41·5-s − 0.408·6-s − 1.53·7-s + 0.353·8-s + 0.333·9-s − 1.00·10-s + 0.718·11-s − 0.288·12-s + 0.0815·13-s − 1.08·14-s + 0.818·15-s + 0.250·16-s − 1.16·17-s + 0.235·18-s − 0.931·19-s − 0.708·20-s + 0.885·21-s + 0.508·22-s + 0.208·23-s − 0.204·24-s + 1.01·25-s + 0.0576·26-s − 0.192·27-s − 0.767·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8744526808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8744526808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 3.17T + 5T^{2} \) |
| 7 | \( 1 + 4.05T + 7T^{2} \) |
| 11 | \( 1 - 2.38T + 11T^{2} \) |
| 13 | \( 1 - 0.293T + 13T^{2} \) |
| 17 | \( 1 + 4.81T + 17T^{2} \) |
| 19 | \( 1 + 4.05T + 19T^{2} \) |
| 31 | \( 1 + 9.51T + 31T^{2} \) |
| 37 | \( 1 + 2.46T + 37T^{2} \) |
| 41 | \( 1 - 0.0864T + 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 - 2.56T + 47T^{2} \) |
| 53 | \( 1 - 3.57T + 53T^{2} \) |
| 59 | \( 1 + 7.42T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 7.08T + 67T^{2} \) |
| 71 | \( 1 - 7.95T + 71T^{2} \) |
| 73 | \( 1 - 14.6T + 73T^{2} \) |
| 79 | \( 1 + 1.22T + 79T^{2} \) |
| 83 | \( 1 + 2.75T + 83T^{2} \) |
| 89 | \( 1 + 0.550T + 89T^{2} \) |
| 97 | \( 1 - 8.84T + 97T^{2} \) |
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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
−8.437405499356499462825116740930, −7.39243834283011514141296186405, −6.73237368766568097684742373475, −6.46420108744802490144041422396, −5.45959878062334545544742199747, −4.48602233913895083676933775648, −3.81944566992992054538397663128, −3.40511996940753958010081109851, −2.15098113762045182370763898782, −0.46977906764429856927438918322,
0.46977906764429856927438918322, 2.15098113762045182370763898782, 3.40511996940753958010081109851, 3.81944566992992054538397663128, 4.48602233913895083676933775648, 5.45959878062334545544742199747, 6.46420108744802490144041422396, 6.73237368766568097684742373475, 7.39243834283011514141296186405, 8.437405499356499462825116740930
