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
−9.625466121805053914292817388904, −9.153721124870136298708456709501, −8.116163480371831850471831394649, −7.62419628857469780160022211587, −6.73520123952445831385073887559, −5.57179115103164990952497771898, −4.50979625974702060744035412558, −3.51924757861951291778317206048, −2.23364191758854302351612996856, −1.07933469548806917265494542448,
1.28465983954969061802517856109, 1.89111067719502284716081261750, 3.67845156231078820560249940694, 4.74824135183312926476497705344, 5.26336996193282523104732682249, 6.68569217162208711731079156414, 7.67673010142092776500119997379, 8.534417092223683913345746973762, 8.788017548836811423346080122899, 9.612767448230672624448912802166