L(s) = 1 | − 3.25·3-s − 2.27·5-s − 4.28i·7-s + 7.58·9-s + i·11-s + 2.82i·13-s + 7.39·15-s − 5.42·17-s + (−4.27 + 0.826i)19-s + 13.9i·21-s − 8.87i·23-s + 0.169·25-s − 14.9·27-s − 5.91i·29-s + 5.48·31-s + ⋯ |
L(s) = 1 | − 1.87·3-s − 1.01·5-s − 1.61i·7-s + 2.52·9-s + 0.301i·11-s + 0.783i·13-s + 1.90·15-s − 1.31·17-s + (−0.981 + 0.189i)19-s + 3.03i·21-s − 1.85i·23-s + 0.0338·25-s − 2.86·27-s − 1.09i·29-s + 0.985·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.06910576641\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06910576641\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - iT \) |
| 19 | \( 1 + (4.27 - 0.826i)T \) |
good | 3 | \( 1 + 3.25T + 3T^{2} \) |
| 5 | \( 1 + 2.27T + 5T^{2} \) |
| 7 | \( 1 + 4.28iT - 7T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 + 5.42T + 17T^{2} \) |
| 23 | \( 1 + 8.87iT - 23T^{2} \) |
| 29 | \( 1 + 5.91iT - 29T^{2} \) |
| 31 | \( 1 - 5.48T + 31T^{2} \) |
| 37 | \( 1 + 3.25iT - 37T^{2} \) |
| 41 | \( 1 - 2.92iT - 41T^{2} \) |
| 43 | \( 1 + 9.28iT - 43T^{2} \) |
| 47 | \( 1 + 5.09iT - 47T^{2} \) |
| 53 | \( 1 + 3.16iT - 53T^{2} \) |
| 59 | \( 1 - 1.30T + 59T^{2} \) |
| 61 | \( 1 - 1.77T + 61T^{2} \) |
| 67 | \( 1 + 8.89T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 5.70T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 + 14.9iT - 83T^{2} \) |
| 89 | \( 1 + 18.0iT - 89T^{2} \) |
| 97 | \( 1 - 14.6iT - 97T^{2} \) |
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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
−7.83862033362544639038001549707, −6.96008093019622096269412387559, −6.75899717701067406147820964807, −6.01761391472218047522401925720, −4.62438419633564003138349876480, −4.39596443374471378359212160268, −3.94946183339123993067478924585, −2.00394841078946402544645504085, −0.60173286025156759294585305971, −0.05158145100042320016333143287,
1.40958688616565043274417771073, 2.76475226704350873505233957454, 3.97262182539143848029558989592, 4.80405832790401849036278361428, 5.42160963779675762610860376786, 6.08352760427470221738244714807, 6.66503965421320249064119583638, 7.58004691587833534625501896662, 8.361953169312776899363841145891, 9.150806533998677104858013790938