| L(s) = 1 | + 1.90·2-s − 1.90·3-s + 1.61·4-s − 3.23·5-s − 3.61·6-s + 0.236·7-s − 0.726·8-s + 0.618·9-s − 6.15·10-s − 1.38·11-s − 3.07·12-s + 0.726·13-s + 0.449·14-s + 6.15·15-s − 4.61·16-s − 6.47·17-s + 1.17·18-s − 5.23·20-s − 0.449·21-s − 2.62·22-s − 0.381·23-s + 1.38·24-s + 5.47·25-s + 1.38·26-s + 4.53·27-s + 0.381·28-s + 4.25·29-s + ⋯ |
| L(s) = 1 | + 1.34·2-s − 1.09·3-s + 0.809·4-s − 1.44·5-s − 1.47·6-s + 0.0892·7-s − 0.256·8-s + 0.206·9-s − 1.94·10-s − 0.416·11-s − 0.888·12-s + 0.201·13-s + 0.120·14-s + 1.58·15-s − 1.15·16-s − 1.56·17-s + 0.277·18-s − 1.17·20-s − 0.0979·21-s − 0.560·22-s − 0.0796·23-s + 0.282·24-s + 1.09·25-s + 0.271·26-s + 0.871·27-s + 0.0721·28-s + 0.789·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 \) |
| good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 3 | \( 1 + 1.90T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 0.236T + 7T^{2} \) |
| 11 | \( 1 + 1.38T + 11T^{2} \) |
| 13 | \( 1 - 0.726T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 23 | \( 1 + 0.381T + 23T^{2} \) |
| 29 | \( 1 - 4.25T + 29T^{2} \) |
| 31 | \( 1 + 1.90T + 31T^{2} \) |
| 37 | \( 1 - 6.60T + 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 - 9.32T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 4.97T + 53T^{2} \) |
| 59 | \( 1 + 8.78T + 59T^{2} \) |
| 61 | \( 1 + 9.47T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 - 4.70T + 79T^{2} \) |
| 83 | \( 1 + 3.23T + 83T^{2} \) |
| 89 | \( 1 + 7.43T + 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
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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
−11.28660184975313400514757657409, −10.77245053048310236871887740121, −9.031000355876771090634653114206, −7.924725248811339056883891938720, −6.74356042380978968815750660623, −5.94626396274281941551987069673, −4.76219929660358618501908430676, −4.26583544710456055851021320368, −2.93933289716220419844977986978, 0,
2.93933289716220419844977986978, 4.26583544710456055851021320368, 4.76219929660358618501908430676, 5.94626396274281941551987069673, 6.74356042380978968815750660623, 7.924725248811339056883891938720, 9.031000355876771090634653114206, 10.77245053048310236871887740121, 11.28660184975313400514757657409
