L(s) = 1 | − 2-s + 3-s + 4-s − 0.975·5-s − 6-s − 3.46·7-s − 8-s + 9-s + 0.975·10-s − 0.271·11-s + 12-s + 13-s + 3.46·14-s − 0.975·15-s + 16-s + 5.01·17-s − 18-s + 1.63·19-s − 0.975·20-s − 3.46·21-s + 0.271·22-s + 4.90·23-s − 24-s − 4.04·25-s − 26-s + 27-s − 3.46·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.436·5-s − 0.408·6-s − 1.30·7-s − 0.353·8-s + 0.333·9-s + 0.308·10-s − 0.0817·11-s + 0.288·12-s + 0.277·13-s + 0.926·14-s − 0.251·15-s + 0.250·16-s + 1.21·17-s − 0.235·18-s + 0.375·19-s − 0.218·20-s − 0.756·21-s + 0.0578·22-s + 1.02·23-s − 0.204·24-s − 0.809·25-s − 0.196·26-s + 0.192·27-s − 0.654·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.206199681\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.206199681\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 0.975T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 0.271T + 11T^{2} \) |
| 17 | \( 1 - 5.01T + 17T^{2} \) |
| 19 | \( 1 - 1.63T + 19T^{2} \) |
| 23 | \( 1 - 4.90T + 23T^{2} \) |
| 29 | \( 1 + 6.97T + 29T^{2} \) |
| 31 | \( 1 - 4.30T + 31T^{2} \) |
| 37 | \( 1 + 6.45T + 37T^{2} \) |
| 41 | \( 1 + 6.97T + 41T^{2} \) |
| 43 | \( 1 - 1.56T + 43T^{2} \) |
| 47 | \( 1 - 3.44T + 47T^{2} \) |
| 53 | \( 1 - 4.10T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 2.73T + 67T^{2} \) |
| 71 | \( 1 + 5.88T + 71T^{2} \) |
| 73 | \( 1 - 3.77T + 73T^{2} \) |
| 79 | \( 1 - 8.97T + 79T^{2} \) |
| 83 | \( 1 + 9.85T + 83T^{2} \) |
| 89 | \( 1 - 9.07T + 89T^{2} \) |
| 97 | \( 1 + 4.57T + 97T^{2} \) |
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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
−7.84243013070443420464001616155, −7.26129045207585723566621092915, −6.71164940471896481151139292847, −5.87971486055681270664258487009, −5.17356473855594042475249790950, −3.89003200694432762194815333208, −3.39832604068264085637390531901, −2.79401979790562737811814081951, −1.67826166350582869230390053057, −0.58738674129857564346682551079,
0.58738674129857564346682551079, 1.67826166350582869230390053057, 2.79401979790562737811814081951, 3.39832604068264085637390531901, 3.89003200694432762194815333208, 5.17356473855594042475249790950, 5.87971486055681270664258487009, 6.71164940471896481151139292847, 7.26129045207585723566621092915, 7.84243013070443420464001616155