L(s) = 1 | + (−1.26 − 0.622i)2-s + (1.22 + 1.58i)4-s − 3.44·7-s + (−0.570 − 2.77i)8-s + 5.54i·11-s − 3.87i·13-s + (4.38 + 2.14i)14-s + (−0.999 + 3.87i)16-s + 6.22·17-s − 7.03i·19-s + (3.44 − 7.03i)22-s − 1.14·23-s + (−2.41 + 4.91i)26-s + (−4.22 − 5.45i)28-s + 3.05i·29-s + ⋯ |
L(s) = 1 | + (−0.897 − 0.440i)2-s + (0.612 + 0.790i)4-s − 1.30·7-s + (−0.201 − 0.979i)8-s + 1.67i·11-s − 1.07i·13-s + (1.17 + 0.573i)14-s + (−0.249 + 0.968i)16-s + 1.50·17-s − 1.61i·19-s + (0.735 − 1.49i)22-s − 0.238·23-s + (−0.472 + 0.964i)26-s + (−0.798 − 1.03i)28-s + 0.566i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.201 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4530678599\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4530678599\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.26 + 0.622i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 3.44T + 7T^{2} \) |
| 11 | \( 1 - 5.54iT - 11T^{2} \) |
| 13 | \( 1 + 3.87iT - 13T^{2} \) |
| 17 | \( 1 - 6.22T + 17T^{2} \) |
| 19 | \( 1 + 7.03iT - 19T^{2} \) |
| 23 | \( 1 + 1.14T + 23T^{2} \) |
| 29 | \( 1 - 3.05iT - 29T^{2} \) |
| 31 | \( 1 + 1.44T + 31T^{2} \) |
| 37 | \( 1 - 6.32iT - 37T^{2} \) |
| 41 | \( 1 - 3.93T + 41T^{2} \) |
| 43 | \( 1 + 0.710iT - 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 - 8.03iT - 53T^{2} \) |
| 59 | \( 1 - 4.98iT - 59T^{2} \) |
| 61 | \( 1 - 10.1iT - 61T^{2} \) |
| 67 | \( 1 - 5.61iT - 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 6.89T + 73T^{2} \) |
| 79 | \( 1 - 2.89T + 79T^{2} \) |
| 83 | \( 1 - 6.10iT - 83T^{2} \) |
| 89 | \( 1 - 7.87T + 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−9.635869393755327409448462854178, −8.954383921019195855466777910741, −7.889812804122491751958075962908, −7.24004212348964740242603394096, −6.64883578109881506512431785506, −5.56754662665462438472899325533, −4.38790327351304681887450468072, −3.19945879292229262876923508528, −2.67019098492418360290094941190, −1.19894069870310017063058508750,
0.24905098608523480154529344892, 1.61843655185879572596156433056, 3.06872271439367829944387205021, 3.78083086793718529230094171329, 5.44140644598811216118138360964, 6.08048266089185512415807576938, 6.54422257643681420589852804088, 7.66568598259095656447097805010, 8.225887110643029049285137450253, 9.123413035640568518176174575588