L(s) = 1 | + i·2-s + (−1.29 + 1.29i)3-s − 4-s + (−0.390 − 2.20i)5-s + (−1.29 − 1.29i)6-s + (−2.67 + 2.67i)7-s − i·8-s − 0.348i·9-s + (2.20 − 0.390i)10-s − 1.29i·11-s + (1.29 − 1.29i)12-s − 6.92i·13-s + (−2.67 − 2.67i)14-s + (3.35 + 2.34i)15-s + 16-s − 4.85·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.747 + 0.747i)3-s − 0.5·4-s + (−0.174 − 0.984i)5-s + (−0.528 − 0.528i)6-s + (−1.00 + 1.00i)7-s − 0.353i·8-s − 0.116i·9-s + (0.696 − 0.123i)10-s − 0.390i·11-s + (0.373 − 0.373i)12-s − 1.92i·13-s + (−0.713 − 0.713i)14-s + (0.866 + 0.604i)15-s + 0.250·16-s − 1.17·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.188340 - 0.154328i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.188340 - 0.154328i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 + (0.390 + 2.20i)T \) |
| 37 | \( 1 + (0.770 - 6.03i)T \) |
good | 3 | \( 1 + (1.29 - 1.29i)T - 3iT^{2} \) |
| 7 | \( 1 + (2.67 - 2.67i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.29iT - 11T^{2} \) |
| 13 | \( 1 + 6.92iT - 13T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 19 | \( 1 + (-4.69 + 4.69i)T - 19iT^{2} \) |
| 23 | \( 1 - 3.33iT - 23T^{2} \) |
| 29 | \( 1 + (3.73 + 3.73i)T + 29iT^{2} \) |
| 31 | \( 1 + (0.146 - 0.146i)T - 31iT^{2} \) |
| 41 | \( 1 + 6.38iT - 41T^{2} \) |
| 43 | \( 1 + 2.51iT - 43T^{2} \) |
| 47 | \( 1 + (2.93 - 2.93i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.30 + 1.30i)T + 53iT^{2} \) |
| 59 | \( 1 + (4.70 - 4.70i)T - 59iT^{2} \) |
| 61 | \( 1 + (7.81 - 7.81i)T - 61iT^{2} \) |
| 67 | \( 1 + (3.10 + 3.10i)T + 67iT^{2} \) |
| 71 | \( 1 + 5.59T + 71T^{2} \) |
| 73 | \( 1 + (0.134 - 0.134i)T - 73iT^{2} \) |
| 79 | \( 1 + (10.8 - 10.8i)T - 79iT^{2} \) |
| 83 | \( 1 + (8.74 + 8.74i)T + 83iT^{2} \) |
| 89 | \( 1 + (3.83 + 3.83i)T + 89iT^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−11.22269931676098206194762723551, −10.07904705316709307133486113118, −9.297139888398845957515727925279, −8.545437938584340951408709262783, −7.44298557984077863744098136464, −5.88629576372518561914933353709, −5.53651240370616795746655242550, −4.58866017228319490715324785701, −3.11323308663633993505312138952, −0.17634696682800565398427978327,
1.76772786516786713595633535844, 3.40095266931311758373976034537, 4.35159732045288803439054011010, 6.16875532573059074421908815417, 6.81595267622483566580779805180, 7.44891803474294456307464505626, 9.194403217494559960121611346533, 9.934091297374598332483427444167, 10.92853053723540080812101311582, 11.52649709246017781910453435223