L(s) = 1 | + 2-s − 3-s + 4-s + 1.56·5-s − 6-s + 7-s + 8-s + 9-s + 1.56·10-s − 2.56·11-s − 12-s + 14-s − 1.56·15-s + 16-s + 0.123·17-s + 18-s − 2.56·19-s + 1.56·20-s − 21-s − 2.56·22-s − 1.12·23-s − 24-s − 2.56·25-s − 27-s + 28-s − 6.12·29-s − 1.56·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.698·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.493·10-s − 0.772·11-s − 0.288·12-s + 0.267·14-s − 0.403·15-s + 0.250·16-s + 0.0298·17-s + 0.235·18-s − 0.587·19-s + 0.349·20-s − 0.218·21-s − 0.546·22-s − 0.234·23-s − 0.204·24-s − 0.512·25-s − 0.192·27-s + 0.188·28-s − 1.13·29-s − 0.285·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 1.56T + 5T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 17 | \( 1 - 0.123T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 + 1.12T + 23T^{2} \) |
| 29 | \( 1 + 6.12T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2.43T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 0.315T + 47T^{2} \) |
| 53 | \( 1 - 7T + 53T^{2} \) |
| 59 | \( 1 - 5.12T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 9.12T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 2.43T + 73T^{2} \) |
| 79 | \( 1 + 6.56T + 79T^{2} \) |
| 83 | \( 1 + 5.12T + 83T^{2} \) |
| 89 | \( 1 - 3.43T + 89T^{2} \) |
| 97 | \( 1 + 0.876T + 97T^{2} \) |
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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
−7.39576992358733845219197991413, −6.74879298502279568039819330919, −5.97943218964778306981878203807, −5.46036227218859083512602350990, −4.91785793108166490089740037714, −4.09809188634619236656263257782, −3.22969739252261663427991978943, −2.17281720287222007118309200925, −1.59075018939489262340828414174, 0,
1.59075018939489262340828414174, 2.17281720287222007118309200925, 3.22969739252261663427991978943, 4.09809188634619236656263257782, 4.91785793108166490089740037714, 5.46036227218859083512602350990, 5.97943218964778306981878203807, 6.74879298502279568039819330919, 7.39576992358733845219197991413