L(s) = 1 | − 2.65·2-s − 3-s + 5.02·4-s + 2.28·5-s + 2.65·6-s + 3.33·7-s − 8.02·8-s + 9-s − 6.05·10-s + 2.21·11-s − 5.02·12-s − 4.45·13-s − 8.84·14-s − 2.28·15-s + 11.2·16-s − 17-s − 2.65·18-s − 6.11·19-s + 11.4·20-s − 3.33·21-s − 5.86·22-s + 7.18·23-s + 8.02·24-s + 0.215·25-s + 11.8·26-s − 27-s + 16.7·28-s + ⋯ |
L(s) = 1 | − 1.87·2-s − 0.577·3-s + 2.51·4-s + 1.02·5-s + 1.08·6-s + 1.26·7-s − 2.83·8-s + 0.333·9-s − 1.91·10-s + 0.667·11-s − 1.45·12-s − 1.23·13-s − 2.36·14-s − 0.589·15-s + 2.80·16-s − 0.242·17-s − 0.624·18-s − 1.40·19-s + 2.56·20-s − 0.728·21-s − 1.25·22-s + 1.49·23-s + 1.63·24-s + 0.0430·25-s + 2.31·26-s − 0.192·27-s + 3.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 5 | \( 1 - 2.28T + 5T^{2} \) |
| 7 | \( 1 - 3.33T + 7T^{2} \) |
| 11 | \( 1 - 2.21T + 11T^{2} \) |
| 13 | \( 1 + 4.45T + 13T^{2} \) |
| 19 | \( 1 + 6.11T + 19T^{2} \) |
| 23 | \( 1 - 7.18T + 23T^{2} \) |
| 29 | \( 1 - 2.31T + 29T^{2} \) |
| 31 | \( 1 + 5.82T + 31T^{2} \) |
| 37 | \( 1 - 4.11T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 8.08T + 43T^{2} \) |
| 47 | \( 1 - 8.28T + 47T^{2} \) |
| 53 | \( 1 + 3.42T + 53T^{2} \) |
| 59 | \( 1 + 8.53T + 59T^{2} \) |
| 61 | \( 1 + 8.76T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 1.97T + 89T^{2} \) |
| 97 | \( 1 + 5.84T + 97T^{2} \) |
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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
−8.250236361025457868281496797249, −7.39925590364414664601982736418, −6.83902280240004938785851732789, −6.18442065097668985234114219452, −5.28432128558229364714809011269, −4.50303478511774002538835718693, −2.80816656064344629238677542678, −1.83981979454437605061604604967, −1.45156491846560159438838524173, 0,
1.45156491846560159438838524173, 1.83981979454437605061604604967, 2.80816656064344629238677542678, 4.50303478511774002538835718693, 5.28432128558229364714809011269, 6.18442065097668985234114219452, 6.83902280240004938785851732789, 7.39925590364414664601982736418, 8.250236361025457868281496797249