| L(s) = 1 | + 0.139·2-s + 0.399·3-s − 1.98·4-s + 2.13·5-s + 0.0556·6-s + 0.319·7-s − 0.554·8-s − 2.84·9-s + 0.297·10-s + 1.88·11-s − 0.791·12-s − 5.95·13-s + 0.0444·14-s + 0.853·15-s + 3.88·16-s + 2.10·17-s − 0.395·18-s − 0.629·19-s − 4.22·20-s + 0.127·21-s + 0.262·22-s − 8.86·23-s − 0.221·24-s − 0.442·25-s − 0.828·26-s − 2.33·27-s − 0.632·28-s + ⋯ |
| L(s) = 1 | + 0.0984·2-s + 0.230·3-s − 0.990·4-s + 0.954·5-s + 0.0227·6-s + 0.120·7-s − 0.195·8-s − 0.946·9-s + 0.0939·10-s + 0.568·11-s − 0.228·12-s − 1.65·13-s + 0.0118·14-s + 0.220·15-s + 0.971·16-s + 0.509·17-s − 0.0932·18-s − 0.144·19-s − 0.945·20-s + 0.0278·21-s + 0.0560·22-s − 1.84·23-s − 0.0452·24-s − 0.0884·25-s − 0.162·26-s − 0.449·27-s − 0.119·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 1511 | \( 1 + T \) |
| good | 2 | \( 1 - 0.139T + 2T^{2} \) |
| 3 | \( 1 - 0.399T + 3T^{2} \) |
| 5 | \( 1 - 2.13T + 5T^{2} \) |
| 7 | \( 1 - 0.319T + 7T^{2} \) |
| 11 | \( 1 - 1.88T + 11T^{2} \) |
| 13 | \( 1 + 5.95T + 13T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 19 | \( 1 + 0.629T + 19T^{2} \) |
| 23 | \( 1 + 8.86T + 23T^{2} \) |
| 29 | \( 1 + 0.674T + 29T^{2} \) |
| 31 | \( 1 - 0.137T + 31T^{2} \) |
| 37 | \( 1 - 1.11T + 37T^{2} \) |
| 41 | \( 1 - 5.50T + 41T^{2} \) |
| 43 | \( 1 + 2.88T + 43T^{2} \) |
| 47 | \( 1 + 4.82T + 47T^{2} \) |
| 53 | \( 1 + 4.26T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 2.98T + 61T^{2} \) |
| 67 | \( 1 - 8.50T + 67T^{2} \) |
| 71 | \( 1 - 4.66T + 71T^{2} \) |
| 73 | \( 1 - 7.85T + 73T^{2} \) |
| 79 | \( 1 + 8.01T + 79T^{2} \) |
| 83 | \( 1 - 5.56T + 83T^{2} \) |
| 89 | \( 1 + 17.9T + 89T^{2} \) |
| 97 | \( 1 + 9.87T + 97T^{2} \) |
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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
−9.427384107190769461120787872818, −8.249450600111231124619704948133, −7.76182870641839915380089467746, −6.40017124679946848939024251174, −5.68711631675654153699735469085, −4.97381720635642782526872967379, −4.01087055064753965121669454456, −2.88406961902596705565370270955, −1.82238219811652285006777334887, 0,
1.82238219811652285006777334887, 2.88406961902596705565370270955, 4.01087055064753965121669454456, 4.97381720635642782526872967379, 5.68711631675654153699735469085, 6.40017124679946848939024251174, 7.76182870641839915380089467746, 8.249450600111231124619704948133, 9.427384107190769461120787872818
