L(s) = 1 | + 1.73i·3-s − 1.01i·5-s + (2.24 − 6.63i)7-s − 2.99·9-s + 10.2·11-s + 8.95i·13-s + 1.75·15-s − 30.4i·17-s − 16.1i·19-s + (11.4 + 3.88i)21-s + 6.72·23-s + 23.9·25-s − 5.19i·27-s + 30·29-s + 50.1i·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s − 0.202i·5-s + (0.320 − 0.947i)7-s − 0.333·9-s + 0.931·11-s + 0.689i·13-s + 0.117·15-s − 1.78i·17-s − 0.849i·19-s + (0.546 + 0.184i)21-s + 0.292·23-s + 0.958·25-s − 0.192i·27-s + 1.03·29-s + 1.61i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.73234 - 0.285014i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.73234 - 0.285014i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 + (-2.24 + 6.63i)T \) |
good | 5 | \( 1 + 1.01iT - 25T^{2} \) |
| 11 | \( 1 - 10.2T + 121T^{2} \) |
| 13 | \( 1 - 8.95iT - 169T^{2} \) |
| 17 | \( 1 + 30.4iT - 289T^{2} \) |
| 19 | \( 1 + 16.1iT - 361T^{2} \) |
| 23 | \( 1 - 6.72T + 529T^{2} \) |
| 29 | \( 1 - 30T + 841T^{2} \) |
| 31 | \( 1 - 50.1iT - 961T^{2} \) |
| 37 | \( 1 - 30.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 7.10iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 74.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 58.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 70.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 0.492iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 2.86iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 27.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 50.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 70.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 133.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 104. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 144. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 100. iT - 9.40e3T^{2} \) |
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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
−11.28587237716837574053074474800, −10.39361443156224350307210831708, −9.344503610903685762841822841094, −8.761883778408543900747719144067, −7.31531265258797736836642919065, −6.62485200196990228710960117962, −4.96258727193984347166071954541, −4.36566891621440198632863308588, −2.98014205816957632494175202748, −0.977286479791208971426043809030,
1.41182387604886467740317420747, 2.79159721369967763927395270146, 4.24229387441356358832590023218, 5.81756136215479197716631535812, 6.31612049165727874497648426676, 7.74020936889443146935460127269, 8.441495246137126213803432684263, 9.402705996282963899029795751390, 10.61195478671717943857130923189, 11.44064713114502108521423479299