L(s) = 1 | + (1.19 + 1.25i)3-s + (1.61 + 1.54i)5-s + (1.98 − 1.14i)7-s + (−0.139 + 2.99i)9-s + (0.864 + 1.49i)11-s + (−5.20 − 3.00i)13-s + (−0.0131 + 3.87i)15-s − 4.65i·17-s − 0.888·19-s + (3.80 + 1.11i)21-s + (5.17 + 2.98i)23-s + (0.198 + 4.99i)25-s + (−3.92 + 3.40i)27-s + (−2.34 − 4.06i)29-s + (−2.98 + 5.16i)31-s + ⋯ |
L(s) = 1 | + (0.690 + 0.723i)3-s + (0.721 + 0.692i)5-s + (0.748 − 0.432i)7-s + (−0.0464 + 0.998i)9-s + (0.260 + 0.451i)11-s + (−1.44 − 0.832i)13-s + (−0.00339 + 0.999i)15-s − 1.12i·17-s − 0.203·19-s + (0.829 + 0.243i)21-s + (1.07 + 0.623i)23-s + (0.0397 + 0.999i)25-s + (−0.754 + 0.656i)27-s + (−0.435 − 0.754i)29-s + (−0.535 + 0.927i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 - 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.69094 + 0.851810i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.69094 + 0.851810i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.19 - 1.25i)T \) |
| 5 | \( 1 + (-1.61 - 1.54i)T \) |
good | 7 | \( 1 + (-1.98 + 1.14i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.864 - 1.49i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.20 + 3.00i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 4.65iT - 17T^{2} \) |
| 19 | \( 1 + 0.888T + 19T^{2} \) |
| 23 | \( 1 + (-5.17 - 2.98i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.34 + 4.06i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.98 - 5.16i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 8.68iT - 37T^{2} \) |
| 41 | \( 1 + (3.15 - 5.46i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.46 + 0.843i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-10.8 + 6.24i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 2.66iT - 53T^{2} \) |
| 59 | \( 1 + (-1.62 + 2.81i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.08 + 10.5i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.8 + 6.25i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 7.25T + 71T^{2} \) |
| 73 | \( 1 + 1.69iT - 73T^{2} \) |
| 79 | \( 1 + (1.99 + 3.44i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.63 - 2.09i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 2.60T + 89T^{2} \) |
| 97 | \( 1 + (3.46 - 1.99i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.24777645075448872053230118306, −10.55119776911636840641962393762, −9.717843300348272550087496638936, −9.098808726936837880406404071352, −7.63211315060332806612843134001, −7.18789614723197440499180176702, −5.43445062869368891308554723361, −4.68039464667593558278895011166, −3.20931430859298008336777719463, −2.14307726972830063280150179154,
1.51823887383352704876690106938, 2.53766501624749086388585316925, 4.30390741186269254943109269858, 5.47537792734625192335655773566, 6.57009861329077319924206029255, 7.62785309159604948641966521174, 8.733373690866038966955920710467, 9.067716640854842391470408460811, 10.24034623748599074378249844763, 11.55373849575069259023812821753