L(s) = 1 | + 326.·2-s − 2.24e3i·3-s + 7.36e4·4-s + 2.24e5i·5-s − 7.30e5i·6-s + 1.90e6i·7-s + 1.33e7·8-s + 9.32e6·9-s + 7.33e7i·10-s + 3.89e7i·11-s − 1.65e8i·12-s − 1.61e8·13-s + 6.19e8i·14-s + 5.03e8·15-s + 1.93e9·16-s + (1.45e9 − 8.66e8i)17-s + ⋯ |
L(s) = 1 | + 1.80·2-s − 0.591i·3-s + 2.24·4-s + 1.28i·5-s − 1.06i·6-s + 0.872i·7-s + 2.24·8-s + 0.650·9-s + 2.31i·10-s + 0.602i·11-s − 1.32i·12-s − 0.713·13-s + 1.57i·14-s + 0.761·15-s + 1.80·16-s + (0.858 − 0.512i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.858 - 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(8)\) |
\(\approx\) |
\(6.030857737\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.030857737\) |
\(L(\frac{17}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-1.45e9 + 8.66e8i)T \) |
good | 2 | \( 1 - 326.T + 3.27e4T^{2} \) |
| 3 | \( 1 + 2.24e3iT - 1.43e7T^{2} \) |
| 5 | \( 1 - 2.24e5iT - 3.05e10T^{2} \) |
| 7 | \( 1 - 1.90e6iT - 4.74e12T^{2} \) |
| 11 | \( 1 - 3.89e7iT - 4.17e15T^{2} \) |
| 13 | \( 1 + 1.61e8T + 5.11e16T^{2} \) |
| 19 | \( 1 - 1.70e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 2.85e10iT - 2.66e20T^{2} \) |
| 29 | \( 1 + 2.44e10iT - 8.62e21T^{2} \) |
| 31 | \( 1 - 2.75e11iT - 2.34e22T^{2} \) |
| 37 | \( 1 - 5.05e10iT - 3.33e23T^{2} \) |
| 41 | \( 1 + 1.19e12iT - 1.55e24T^{2} \) |
| 43 | \( 1 + 9.60e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + 1.33e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 1.42e13T + 7.31e25T^{2} \) |
| 59 | \( 1 - 3.43e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 3.02e13iT - 6.02e26T^{2} \) |
| 67 | \( 1 + 8.15e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 1.45e13iT - 5.87e27T^{2} \) |
| 73 | \( 1 - 7.56e12iT - 8.90e27T^{2} \) |
| 79 | \( 1 + 2.54e14iT - 2.91e28T^{2} \) |
| 83 | \( 1 - 1.50e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 3.18e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 9.45e14iT - 6.33e29T^{2} \) |
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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
−14.84060902799546180003653737664, −14.15048214469853668095258639636, −12.61782530608958091773669371700, −11.94458308839524855739307058496, −10.34081688486697533497338124752, −7.30162850591679009965555427860, −6.46122436173923843237014929840, −4.90339260835715152284704461738, −3.10013032326691276885144255952, −2.10675694517766282358331810080,
1.30655157747277102605595633127, 3.55705611278455970896714645673, 4.54086785476175580412229710422, 5.58505355873626104733890548921, 7.51776648301407033685987023177, 9.837092501510907203783014984631, 11.49390152049023548652316132944, 12.78934385246879192296509552022, 13.58219417353255317573917155353, 14.96329486189237143862846159820