| L(s) = 1 | − 1.71·2-s + 3-s + 0.948·4-s − 2.53·5-s − 1.71·6-s − 0.257·7-s + 1.80·8-s + 9-s + 4.35·10-s + 1.07·11-s + 0.948·12-s − 13-s + 0.442·14-s − 2.53·15-s − 4.99·16-s + 0.765·17-s − 1.71·18-s − 1.58·19-s − 2.40·20-s − 0.257·21-s − 1.84·22-s + 2.51·23-s + 1.80·24-s + 1.43·25-s + 1.71·26-s + 27-s − 0.244·28-s + ⋯ |
| L(s) = 1 | − 1.21·2-s + 0.577·3-s + 0.474·4-s − 1.13·5-s − 0.701·6-s − 0.0973·7-s + 0.638·8-s + 0.333·9-s + 1.37·10-s + 0.324·11-s + 0.273·12-s − 0.277·13-s + 0.118·14-s − 0.655·15-s − 1.24·16-s + 0.185·17-s − 0.404·18-s − 0.364·19-s − 0.538·20-s − 0.0562·21-s − 0.393·22-s + 0.524·23-s + 0.368·24-s + 0.287·25-s + 0.336·26-s + 0.192·27-s − 0.0462·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 + 1.71T + 2T^{2} \) |
| 5 | \( 1 + 2.53T + 5T^{2} \) |
| 7 | \( 1 + 0.257T + 7T^{2} \) |
| 11 | \( 1 - 1.07T + 11T^{2} \) |
| 17 | \( 1 - 0.765T + 17T^{2} \) |
| 19 | \( 1 + 1.58T + 19T^{2} \) |
| 23 | \( 1 - 2.51T + 23T^{2} \) |
| 29 | \( 1 + 4.62T + 29T^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 37 | \( 1 + 2.30T + 37T^{2} \) |
| 41 | \( 1 - 6.15T + 41T^{2} \) |
| 43 | \( 1 - 9.79T + 43T^{2} \) |
| 47 | \( 1 - 1.20T + 47T^{2} \) |
| 53 | \( 1 + 9.48T + 53T^{2} \) |
| 59 | \( 1 - 4.05T + 59T^{2} \) |
| 61 | \( 1 + 7.58T + 61T^{2} \) |
| 67 | \( 1 - 4.67T + 67T^{2} \) |
| 71 | \( 1 - 2.46T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 + 7.47T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 5.42T + 89T^{2} \) |
| 97 | \( 1 + 3.73T + 97T^{2} \) |
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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
−8.113158120215337616062022910739, −7.57526143409751350089205883450, −7.12601525190477106575554280294, −6.12484733182274112074257395552, −4.83175878700429147079354987931, −4.16916353997598936729623922197, −3.38110255212835920677966678437, −2.28594544801099318191702111640, −1.15566775160125572572985608510, 0,
1.15566775160125572572985608510, 2.28594544801099318191702111640, 3.38110255212835920677966678437, 4.16916353997598936729623922197, 4.83175878700429147079354987931, 6.12484733182274112074257395552, 7.12601525190477106575554280294, 7.57526143409751350089205883450, 8.113158120215337616062022910739
