| L(s) = 1 | + 4·2-s − 14·3-s + 16·4-s − 45·5-s − 56·6-s − 121·7-s + 64·8-s − 47·9-s − 180·10-s − 381·11-s − 224·12-s − 100·13-s − 484·14-s + 630·15-s + 256·16-s + 933·17-s − 188·18-s + 361·19-s − 720·20-s + 1.69e3·21-s − 1.52e3·22-s − 552·23-s − 896·24-s − 1.10e3·25-s − 400·26-s + 4.06e3·27-s − 1.93e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.898·3-s + 1/2·4-s − 0.804·5-s − 0.635·6-s − 0.933·7-s + 0.353·8-s − 0.193·9-s − 0.569·10-s − 0.949·11-s − 0.449·12-s − 0.164·13-s − 0.659·14-s + 0.722·15-s + 1/4·16-s + 0.782·17-s − 0.136·18-s + 0.229·19-s − 0.402·20-s + 0.838·21-s − 0.671·22-s − 0.217·23-s − 0.317·24-s − 0.351·25-s − 0.116·26-s + 1.07·27-s − 0.466·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 19 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 14 T + p^{5} T^{2} \) |
| 5 | \( 1 + 9 p T + p^{5} T^{2} \) |
| 7 | \( 1 + 121 T + p^{5} T^{2} \) |
| 11 | \( 1 + 381 T + p^{5} T^{2} \) |
| 13 | \( 1 + 100 T + p^{5} T^{2} \) |
| 17 | \( 1 - 933 T + p^{5} T^{2} \) |
| 23 | \( 1 + 24 p T + p^{5} T^{2} \) |
| 29 | \( 1 - 2394 T + p^{5} T^{2} \) |
| 31 | \( 1 + 4024 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9182 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2250 T + p^{5} T^{2} \) |
| 43 | \( 1 + 23377 T + p^{5} T^{2} \) |
| 47 | \( 1 + 26595 T + p^{5} T^{2} \) |
| 53 | \( 1 + 16008 T + p^{5} T^{2} \) |
| 59 | \( 1 + 126 T + p^{5} T^{2} \) |
| 61 | \( 1 - 21335 T + p^{5} T^{2} \) |
| 67 | \( 1 + 51760 T + p^{5} T^{2} \) |
| 71 | \( 1 - 8574 T + p^{5} T^{2} \) |
| 73 | \( 1 - 11153 T + p^{5} T^{2} \) |
| 79 | \( 1 + 1660 T + p^{5} T^{2} \) |
| 83 | \( 1 - 95964 T + p^{5} T^{2} \) |
| 89 | \( 1 - 118848 T + p^{5} T^{2} \) |
| 97 | \( 1 + 153760 T + p^{5} T^{2} \) |
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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
−14.80038065181427243896005808791, −13.26591311374447780817568943716, −12.23220107129592515046709090492, −11.34093328042158452640592548627, −10.05172806649055001376178094768, −7.88760755769310827187039688961, −6.37630269466196314649986519017, −5.08911044190535973504531876794, −3.26108353113143881237626703375, 0,
3.26108353113143881237626703375, 5.08911044190535973504531876794, 6.37630269466196314649986519017, 7.88760755769310827187039688961, 10.05172806649055001376178094768, 11.34093328042158452640592548627, 12.23220107129592515046709090492, 13.26591311374447780817568943716, 14.80038065181427243896005808791
