| L(s) = 1 | + (0.725 − 3.93i)2-s − 5.19i·3-s + (−14.9 − 5.70i)4-s + (−20.4 − 3.76i)6-s − 36.7i·7-s + (−33.2 + 54.6i)8-s − 27·9-s − 156. i·11-s + (−29.6 + 77.6i)12-s − 62.1·13-s + (−144. − 26.6i)14-s + (190. + 170. i)16-s − 306.·17-s + (−19.5 + 106. i)18-s − 242. i·19-s + ⋯ |
| L(s) = 1 | + (0.181 − 0.983i)2-s − 0.577i·3-s + (−0.934 − 0.356i)4-s + (−0.567 − 0.104i)6-s − 0.749i·7-s + (−0.520 + 0.854i)8-s − 0.333·9-s − 1.29i·11-s + (−0.205 + 0.539i)12-s − 0.367·13-s + (−0.737 − 0.135i)14-s + (0.745 + 0.666i)16-s − 1.06·17-s + (−0.0604 + 0.327i)18-s − 0.672i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.3988343951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3988343951\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.725 + 3.93i)T \) |
| 3 | \( 1 + 5.19iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 36.7iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 156. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 62.1T + 2.85e4T^{2} \) |
| 17 | \( 1 + 306.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 242. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 571. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 266.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 759. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.21e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.67e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.63e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.37e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.39e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 2.90e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 4.78e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 4.65e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 6.76e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 6.58e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 8.54e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 2.17e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.63e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.57e4T + 8.85e7T^{2} \) |
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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
−10.74230591820090859811555137580, −9.435215834871917859989167313481, −8.625132687208847363739990681441, −7.50172292738488728915519652855, −6.26886846476299552914163944664, −5.08671759158324436020018809328, −3.82257916655712402918464195334, −2.70692386453886444864070070767, −1.27013600004924590593465156357, −0.12144348590407086935102056185,
2.38107909588972444527412326656, 4.05377283701720293447024759301, 4.86228354517374131336222896805, 5.91126052182425660678177783018, 6.94692898712978569943142587981, 8.006894056824770631470477775659, 9.039738102251842704930807408300, 9.651491668108582551866932310458, 10.75839280958236770795874886256, 12.19547189427719645794504095353
