L(s) = 1 | − 1.12·2-s + 3-s − 0.744·4-s + (1.27 − 1.83i)5-s − 1.12·6-s + (2.54 + 0.709i)7-s + 3.07·8-s + 9-s + (−1.42 + 2.05i)10-s + (2.57 + 2.09i)11-s − 0.744·12-s − 0.476i·13-s + (−2.85 − 0.794i)14-s + (1.27 − 1.83i)15-s − 1.95·16-s + 4.13i·17-s + ⋯ |
L(s) = 1 | − 0.792·2-s + 0.577·3-s − 0.372·4-s + (0.569 − 0.822i)5-s − 0.457·6-s + (0.963 + 0.268i)7-s + 1.08·8-s + 0.333·9-s + (−0.451 + 0.651i)10-s + (0.775 + 0.631i)11-s − 0.214·12-s − 0.132i·13-s + (−0.763 − 0.212i)14-s + (0.328 − 0.474i)15-s − 0.489·16-s + 1.00i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.563714501\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.563714501\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + (-1.27 + 1.83i)T \) |
| 7 | \( 1 + (-2.54 - 0.709i)T \) |
| 11 | \( 1 + (-2.57 - 2.09i)T \) |
good | 2 | \( 1 + 1.12T + 2T^{2} \) |
| 13 | \( 1 + 0.476iT - 13T^{2} \) |
| 17 | \( 1 - 4.13iT - 17T^{2} \) |
| 19 | \( 1 - 3.40T + 19T^{2} \) |
| 23 | \( 1 + 2.30iT - 23T^{2} \) |
| 29 | \( 1 - 8.78iT - 29T^{2} \) |
| 31 | \( 1 - 7.12iT - 31T^{2} \) |
| 37 | \( 1 + 4.59iT - 37T^{2} \) |
| 41 | \( 1 + 6.75T + 41T^{2} \) |
| 43 | \( 1 - 4.78T + 43T^{2} \) |
| 47 | \( 1 + 9.14T + 47T^{2} \) |
| 53 | \( 1 + 4.46iT - 53T^{2} \) |
| 59 | \( 1 + 10.6iT - 59T^{2} \) |
| 61 | \( 1 - 2.11T + 61T^{2} \) |
| 67 | \( 1 - 11.4iT - 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 9.29iT - 73T^{2} \) |
| 79 | \( 1 + 0.726iT - 79T^{2} \) |
| 83 | \( 1 + 8.78iT - 83T^{2} \) |
| 89 | \( 1 - 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 4.36T + 97T^{2} \) |
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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
−9.612767448230672624448912802166, −8.788017548836811423346080122899, −8.534417092223683913345746973762, −7.67673010142092776500119997379, −6.68569217162208711731079156414, −5.26336996193282523104732682249, −4.74824135183312926476497705344, −3.67845156231078820560249940694, −1.89111067719502284716081261750, −1.28465983954969061802517856109,
1.07933469548806917265494542448, 2.23364191758854302351612996856, 3.51924757861951291778317206048, 4.50979625974702060744035412558, 5.57179115103164990952497771898, 6.73520123952445831385073887559, 7.62419628857469780160022211587, 8.116163480371831850471831394649, 9.153721124870136298708456709501, 9.625466121805053914292817388904