L(s) = 1 | − 2-s − 3-s + 4-s − 2.24·5-s + 6-s − 4.80·7-s − 8-s + 9-s + 2.24·10-s − 5.91·11-s − 12-s − 13-s + 4.80·14-s + 2.24·15-s + 16-s + 0.672·17-s − 18-s − 6.07·19-s − 2.24·20-s + 4.80·21-s + 5.91·22-s − 1.49·23-s + 24-s + 0.0270·25-s + 26-s − 27-s − 4.80·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.00·5-s + 0.408·6-s − 1.81·7-s − 0.353·8-s + 0.333·9-s + 0.709·10-s − 1.78·11-s − 0.288·12-s − 0.277·13-s + 1.28·14-s + 0.578·15-s + 0.250·16-s + 0.163·17-s − 0.235·18-s − 1.39·19-s − 0.501·20-s + 1.04·21-s + 1.26·22-s − 0.311·23-s + 0.204·24-s + 0.00540·25-s + 0.196·26-s − 0.192·27-s − 0.908·28-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 | \( 1 + 2.24T + 5T^{2} \) |
| 7 | \( 1 + 4.80T + 7T^{2} \) |
| 11 | \( 1 + 5.91T + 11T^{2} \) |
| 17 | \( 1 - 0.672T + 17T^{2} \) |
| 19 | \( 1 + 6.07T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 - 4.90T + 29T^{2} \) |
| 31 | \( 1 + 0.627T + 31T^{2} \) |
| 37 | \( 1 - 9.51T + 37T^{2} \) |
| 41 | \( 1 - 4.06T + 41T^{2} \) |
| 43 | \( 1 - 8.49T + 43T^{2} \) |
| 47 | \( 1 - 3.45T + 47T^{2} \) |
| 53 | \( 1 - 7.99T + 53T^{2} \) |
| 59 | \( 1 - 4.41T + 59T^{2} \) |
| 61 | \( 1 + 5.58T + 61T^{2} \) |
| 67 | \( 1 + 0.920T + 67T^{2} \) |
| 71 | \( 1 + 1.63T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 6.26T + 79T^{2} \) |
| 83 | \( 1 + 5.91T + 83T^{2} \) |
| 89 | \( 1 - 8.71T + 89T^{2} \) |
| 97 | \( 1 + 4.09T + 97T^{2} \) |
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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
−7.45104492673049295233442415851, −6.99178823034219815343357071351, −6.04578975265321020891992859833, −5.78177605654444875082346768264, −4.53901021289214809994491845581, −3.92362781766504775664671779288, −2.86440448330526841417721886344, −2.44542128339835244499891040466, −0.64238703580943644630865332956, 0,
0.64238703580943644630865332956, 2.44542128339835244499891040466, 2.86440448330526841417721886344, 3.92362781766504775664671779288, 4.53901021289214809994491845581, 5.78177605654444875082346768264, 6.04578975265321020891992859833, 6.99178823034219815343357071351, 7.45104492673049295233442415851