L(s) = 1 | − 1.41i·2-s + 5.56i·3-s − 2.00·4-s + (−0.637 + 4.95i)5-s + 7.87·6-s + 10.0·7-s + 2.82i·8-s − 22.0·9-s + (7.01 + 0.901i)10-s + 9.80i·11-s − 11.1i·12-s − 14.3i·13-s − 14.1i·14-s + (−27.6 − 3.54i)15-s + 4.00·16-s − 17.3·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 1.85i·3-s − 0.500·4-s + (−0.127 + 0.991i)5-s + 1.31·6-s + 1.43·7-s + 0.353i·8-s − 2.44·9-s + (0.701 + 0.0901i)10-s + 0.891i·11-s − 0.928i·12-s − 1.10i·13-s − 1.01i·14-s + (−1.84 − 0.236i)15-s + 0.250·16-s − 1.02·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.663046 + 1.19761i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.663046 + 1.19761i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 + (0.637 - 4.95i)T \) |
| 23 | \( 1 + (-14.5 - 17.7i)T \) |
good | 3 | \( 1 - 5.56iT - 9T^{2} \) |
| 7 | \( 1 - 10.0T + 49T^{2} \) |
| 11 | \( 1 - 9.80iT - 121T^{2} \) |
| 13 | \( 1 + 14.3iT - 169T^{2} \) |
| 17 | \( 1 + 17.3T + 289T^{2} \) |
| 19 | \( 1 - 8.38iT - 361T^{2} \) |
| 29 | \( 1 + 26.9T + 841T^{2} \) |
| 31 | \( 1 - 4.24T + 961T^{2} \) |
| 37 | \( 1 - 73.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 19.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + 18.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 10.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 90.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 92.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 36.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 41.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 22.7T + 5.04e3T^{2} \) |
| 73 | \( 1 - 45.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 65.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 4.65T + 6.88e3T^{2} \) |
| 89 | \( 1 - 121. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 99.8T + 9.40e3T^{2} \) |
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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
−11.62710314884741728937706014028, −11.20990656869685048342026780724, −10.43625248256104994688035222979, −9.800993236401415135620844983577, −8.685269018805341581013453302080, −7.62156634968044982784577436841, −5.61281357340239730220697102910, −4.66134908993060943504094278372, −3.76316636425355058123967761058, −2.46730352479968413461126696465,
0.76678084660087687831807371338, 2.03660752788460830196884817225, 4.52276095389575289150955833246, 5.64624429960127996741005021226, 6.71863620927169137383612330397, 7.65242841118974245114682420443, 8.525401024273148006620961852956, 8.900868342064776228234592234488, 11.33597784879463120157699036117, 11.62846861809566184524864339450