L(s) = 1 | − i·2-s − 2i·3-s − 4-s − 2·6-s − 5i·7-s + i·8-s − 9-s − 3·11-s + 2i·12-s + i·13-s − 5·14-s + 16-s − 3i·17-s + i·18-s + 4·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 1.15i·3-s − 0.5·4-s − 0.816·6-s − 1.88i·7-s + 0.353i·8-s − 0.333·9-s − 0.904·11-s + 0.577i·12-s + 0.277i·13-s − 1.33·14-s + 0.250·16-s − 0.727i·17-s + 0.235i·18-s + 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.262689 + 1.11276i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.262689 + 1.11276i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + 5iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 17 | \( 1 + 3iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 9T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 - 9T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 5iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 15iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−10.08539463281136176625728688899, −9.493397950989875749617801935385, −7.919052387387748458797192749329, −7.52612911527824084085126204832, −6.80236211466351423378386627784, −5.39995479491548125156213249436, −4.24721484283981736705875816276, −3.17247211195596924163512417828, −1.71702188709885502508179621959, −0.63195364148332238163188775522,
2.44979204919754918989041777826, 3.65565428087899144511564961449, 4.98463963356977790824644265893, 5.42171551094545322639681337571, 6.33748866561646631587654187170, 7.73970694839697808678448658724, 8.584494492550617479988325055388, 9.218207064177277851178432758955, 10.01381352146786294455727040879, 10.80263401229139693958530363711