| L(s) = 1 | − 2.21·2-s + (1.64 − 0.539i)3-s + 2.90·4-s + i·5-s + (−3.64 + 1.19i)6-s − i·7-s − 2.00·8-s + (2.41 − 1.77i)9-s − 2.21i·10-s + (−3.14 + 1.06i)11-s + (4.77 − 1.56i)12-s + 2.21i·13-s + 2.21i·14-s + (0.539 + 1.64i)15-s − 1.37·16-s + 2.72·17-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + (0.950 − 0.311i)3-s + 1.45·4-s + 0.447i·5-s + (−1.48 + 0.487i)6-s − 0.377i·7-s − 0.707·8-s + (0.806 − 0.591i)9-s − 0.700i·10-s + (−0.947 + 0.320i)11-s + (1.37 − 0.452i)12-s + 0.613i·13-s + 0.591i·14-s + (0.139 + 0.424i)15-s − 0.343·16-s + 0.660·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9827459304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9827459304\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.64 + 0.539i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (3.14 - 1.06i)T \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 13 | \( 1 - 2.21iT - 13T^{2} \) |
| 17 | \( 1 - 2.72T + 17T^{2} \) |
| 19 | \( 1 - 2.83iT - 19T^{2} \) |
| 23 | \( 1 - 5.72iT - 23T^{2} \) |
| 29 | \( 1 - 1.05T + 29T^{2} \) |
| 31 | \( 1 - 7.54T + 31T^{2} \) |
| 37 | \( 1 + 5.29T + 37T^{2} \) |
| 41 | \( 1 - 9.83T + 41T^{2} \) |
| 43 | \( 1 - 5.48iT - 43T^{2} \) |
| 47 | \( 1 - 4.94iT - 47T^{2} \) |
| 53 | \( 1 - 10.4iT - 53T^{2} \) |
| 59 | \( 1 + 14.4iT - 59T^{2} \) |
| 61 | \( 1 - 4.24iT - 61T^{2} \) |
| 67 | \( 1 - 2.41T + 67T^{2} \) |
| 71 | \( 1 + 2.12iT - 71T^{2} \) |
| 73 | \( 1 + 3.61iT - 73T^{2} \) |
| 79 | \( 1 - 1.37iT - 79T^{2} \) |
| 83 | \( 1 + 1.18T + 83T^{2} \) |
| 89 | \( 1 + 0.596iT - 89T^{2} \) |
| 97 | \( 1 + 3.76T + 97T^{2} \) |
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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
−9.777365542153353951890852642108, −9.130632761944982935205702514449, −8.096013075964255985402726740265, −7.71864892036560438932306036022, −7.08811231979640080724808403395, −6.12426211120195589113630419082, −4.52724358275015579030089532251, −3.26581446411353637031908047540, −2.25175174888274798614217170093, −1.20432377732305626682361976534,
0.72858080987641241315067050530, 2.21487624051166728005443791403, 2.99257580936425316283063262510, 4.47602975467477697004758893879, 5.48085638773938134463267630830, 6.82841394875202012518196011564, 7.72124362931253899530758892182, 8.365035967534506287379737380126, 8.716620438200231496627222728957, 9.606733660968340492052493338109
