L(s) = 1 | − 1.41·5-s − 1.41i·11-s + i·13-s + 1.00·25-s − 1.41i·41-s − 2·43-s − 1.41·47-s − 49-s + 2.00i·55-s − 1.41i·59-s − 2i·61-s − 1.41i·65-s − 1.41·71-s − 2i·79-s + 1.41i·83-s + ⋯ |
L(s) = 1 | − 1.41·5-s − 1.41i·11-s + i·13-s + 1.00·25-s − 1.41i·41-s − 2·43-s − 1.41·47-s − 49-s + 2.00i·55-s − 1.41i·59-s − 2i·61-s − 1.41i·65-s − 1.41·71-s − 2i·79-s + 1.41i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4035600763\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4035600763\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 5 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + 1.41iT - T^{2} \) |
| 43 | \( 1 + 2T + T^{2} \) |
| 47 | \( 1 + 1.41T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 + 2iT - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + 1.41T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 - 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.259272526010509065886764315779, −7.916457056309603293895373732166, −6.87775997071742131338237171996, −6.40666850339013434706927517219, −5.29869567702182160911535182821, −4.53243875330064760256872348327, −3.61567961242528486957376533627, −3.22032935971700129934475445052, −1.75662254459451993643855768097, −0.23638912551924799575035268591,
1.44957650514385746260008170959, 2.80244367298260897693151654602, 3.55064850449872361463417715247, 4.48851162588584534685624536520, 4.92944194213764110226708443291, 6.05539036938399166488887591250, 7.02103339107924165075480165321, 7.49919155086612550457125551281, 8.158696444600926723962639049710, 8.728481323333251503511201870831