| L(s) = 1 | + (0.870 − 0.492i)2-s + (−0.520 − 0.488i)3-s + (0.515 − 0.856i)4-s + (2.16 + 0.0796i)5-s + (−0.693 − 0.168i)6-s + (0.476 + 2.43i)7-s + (0.0275 − 0.999i)8-s + (−0.160 − 2.48i)9-s + (1.92 − 0.996i)10-s + (2.93 + 3.98i)11-s + (−0.686 + 0.194i)12-s + (−1.41 − 0.606i)13-s + (1.61 + 1.88i)14-s + (−1.08 − 1.09i)15-s + (−0.467 − 0.883i)16-s + (6.55 − 0.120i)17-s + ⋯ |
| L(s) = 1 | + (0.615 − 0.347i)2-s + (−0.300 − 0.281i)3-s + (0.257 − 0.428i)4-s + (0.969 + 0.0356i)5-s + (−0.283 − 0.0689i)6-s + (0.179 + 0.921i)7-s + (0.00974 − 0.353i)8-s + (−0.0533 − 0.828i)9-s + (0.608 − 0.315i)10-s + (0.883 + 1.20i)11-s + (−0.198 + 0.0560i)12-s + (−0.393 − 0.168i)13-s + (0.431 + 0.504i)14-s + (−0.281 − 0.283i)15-s + (−0.116 − 0.220i)16-s + (1.58 − 0.0292i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.32013 - 0.738413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.32013 - 0.738413i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.870 + 0.492i)T \) |
| 19 | \( 1 + (4.10 + 1.46i)T \) |
| good | 3 | \( 1 + (0.520 + 0.488i)T + (0.192 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-2.16 - 0.0796i)T + (4.98 + 0.367i)T^{2} \) |
| 7 | \( 1 + (-0.476 - 2.43i)T + (-6.48 + 2.63i)T^{2} \) |
| 11 | \( 1 + (-2.93 - 3.98i)T + (-3.28 + 10.4i)T^{2} \) |
| 13 | \( 1 + (1.41 + 0.606i)T + (8.97 + 9.40i)T^{2} \) |
| 17 | \( 1 + (-6.55 + 0.120i)T + (16.9 - 0.624i)T^{2} \) |
| 23 | \( 1 + (-8.16 - 2.47i)T + (19.1 + 12.7i)T^{2} \) |
| 29 | \( 1 + (3.67 + 2.97i)T + (6.08 + 28.3i)T^{2} \) |
| 31 | \( 1 + (1.99 + 5.32i)T + (-23.3 + 20.3i)T^{2} \) |
| 37 | \( 1 + (2.41 - 5.50i)T + (-25.0 - 27.2i)T^{2} \) |
| 41 | \( 1 + (-3.17 - 0.292i)T + (40.3 + 7.49i)T^{2} \) |
| 43 | \( 1 + (0.391 + 8.52i)T + (-42.8 + 3.94i)T^{2} \) |
| 47 | \( 1 + (3.13 - 3.28i)T + (-2.15 - 46.9i)T^{2} \) |
| 53 | \( 1 + (3.26 + 11.1i)T + (-44.6 + 28.5i)T^{2} \) |
| 59 | \( 1 + (-3.69 - 8.02i)T + (-38.3 + 44.8i)T^{2} \) |
| 61 | \( 1 + (-4.32 + 0.886i)T + (56.0 - 23.9i)T^{2} \) |
| 67 | \( 1 + (10.1 - 1.50i)T + (64.1 - 19.4i)T^{2} \) |
| 71 | \( 1 + (1.69 + 0.346i)T + (65.2 + 27.9i)T^{2} \) |
| 73 | \( 1 + (3.77 - 6.81i)T + (-38.7 - 61.8i)T^{2} \) |
| 79 | \( 1 + (-8.11 + 5.19i)T + (33.0 - 71.7i)T^{2} \) |
| 83 | \( 1 + (0.0753 - 0.543i)T + (-79.8 - 22.5i)T^{2} \) |
| 89 | \( 1 + (6.17 + 11.1i)T + (-47.3 + 75.3i)T^{2} \) |
| 97 | \( 1 + (8.11 + 1.20i)T + (92.8 + 28.1i)T^{2} \) |
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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
−10.22180540738373872537786591890, −9.550316938292765257603754670867, −8.974213262666285159552547555987, −7.42584450573446444889036693679, −6.56047755225042734625374594214, −5.75125916706575515605799332906, −5.08203036278518437582523258028, −3.75377549766719143351061214490, −2.46314489921252386389195494089, −1.44002612439947133223778304898,
1.45930789534574752099032339791, 3.06996101339254300983706077893, 4.16711103413871753573709884755, 5.22175957179250297389345250115, 5.85237321790090318264433817629, 6.82549931338085328717061408720, 7.73171917385833471433105614934, 8.753702447124374627113377943092, 9.721192429312329635418431546460, 10.77889744456177106774279299821
