L(s) = 1 | + 2-s − 3-s + 4-s + 2.47·5-s − 6-s − 2.89·7-s + 8-s + 9-s + 2.47·10-s + 2.90·11-s − 12-s − 13-s − 2.89·14-s − 2.47·15-s + 16-s + 0.600·17-s + 18-s − 5.25·19-s + 2.47·20-s + 2.89·21-s + 2.90·22-s + 3.98·23-s − 24-s + 1.14·25-s − 26-s − 27-s − 2.89·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.10·5-s − 0.408·6-s − 1.09·7-s + 0.353·8-s + 0.333·9-s + 0.784·10-s + 0.874·11-s − 0.288·12-s − 0.277·13-s − 0.772·14-s − 0.640·15-s + 0.250·16-s + 0.145·17-s + 0.235·18-s − 1.20·19-s + 0.554·20-s + 0.630·21-s + 0.618·22-s + 0.830·23-s − 0.204·24-s + 0.229·25-s − 0.196·26-s − 0.192·27-s − 0.546·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.47T + 5T^{2} \) |
| 7 | \( 1 + 2.89T + 7T^{2} \) |
| 11 | \( 1 - 2.90T + 11T^{2} \) |
| 17 | \( 1 - 0.600T + 17T^{2} \) |
| 19 | \( 1 + 5.25T + 19T^{2} \) |
| 23 | \( 1 - 3.98T + 23T^{2} \) |
| 29 | \( 1 + 3.18T + 29T^{2} \) |
| 31 | \( 1 + 3.32T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 2.64T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 + 3.86T + 53T^{2} \) |
| 59 | \( 1 - 1.08T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 7.96T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 - 4.57T + 79T^{2} \) |
| 83 | \( 1 + 2.92T + 83T^{2} \) |
| 89 | \( 1 - 8.74T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−6.97549499856822328700980004319, −6.55972080164263361047168586372, −6.25083115057123868413070849649, −5.33139257522734980110794351151, −4.93363292428955604259909063645, −3.79718918383601325339839643109, −3.30966417431669190776591051105, −2.17149545501427469266534759467, −1.51868004871217650689192651200, 0,
1.51868004871217650689192651200, 2.17149545501427469266534759467, 3.30966417431669190776591051105, 3.79718918383601325339839643109, 4.93363292428955604259909063645, 5.33139257522734980110794351151, 6.25083115057123868413070849649, 6.55972080164263361047168586372, 6.97549499856822328700980004319