L(s) = 1 | + 0.633i·2-s + (−0.915 − 1.47i)3-s + 1.59·4-s + 0.119·5-s + (0.931 − 0.579i)6-s + (2.16 + 1.52i)7-s + 2.28i·8-s + (−1.32 + 2.69i)9-s + 0.0758i·10-s − 4.57i·11-s + (−1.46 − 2.35i)12-s − i·13-s + (−0.963 + 1.37i)14-s + (−0.109 − 0.176i)15-s + 1.75·16-s + 6.89·17-s + ⋯ |
L(s) = 1 | + 0.448i·2-s + (−0.528 − 0.848i)3-s + 0.799·4-s + 0.0535·5-s + (0.380 − 0.236i)6-s + (0.818 + 0.574i)7-s + 0.806i·8-s + (−0.441 + 0.897i)9-s + 0.0239i·10-s − 1.37i·11-s + (−0.422 − 0.678i)12-s − 0.277i·13-s + (−0.257 + 0.366i)14-s + (−0.0282 − 0.0454i)15-s + 0.438·16-s + 1.67·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41607 - 0.0392135i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41607 - 0.0392135i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.915 + 1.47i)T \) |
| 7 | \( 1 + (-2.16 - 1.52i)T \) |
| 13 | \( 1 + iT \) |
good | 2 | \( 1 - 0.633iT - 2T^{2} \) |
| 5 | \( 1 - 0.119T + 5T^{2} \) |
| 11 | \( 1 + 4.57iT - 11T^{2} \) |
| 17 | \( 1 - 6.89T + 17T^{2} \) |
| 19 | \( 1 - 0.125iT - 19T^{2} \) |
| 23 | \( 1 + 1.84iT - 23T^{2} \) |
| 29 | \( 1 - 3.43iT - 29T^{2} \) |
| 31 | \( 1 + 0.162iT - 31T^{2} \) |
| 37 | \( 1 + 6.26T + 37T^{2} \) |
| 41 | \( 1 + 7.36T + 41T^{2} \) |
| 43 | \( 1 - 1.26T + 43T^{2} \) |
| 47 | \( 1 + 2.57T + 47T^{2} \) |
| 53 | \( 1 + 6.10iT - 53T^{2} \) |
| 59 | \( 1 + 7.02T + 59T^{2} \) |
| 61 | \( 1 - 2.09iT - 61T^{2} \) |
| 67 | \( 1 + 1.47T + 67T^{2} \) |
| 71 | \( 1 - 5.86iT - 71T^{2} \) |
| 73 | \( 1 - 15.5iT - 73T^{2} \) |
| 79 | \( 1 + 9.95T + 79T^{2} \) |
| 83 | \( 1 - 7.28T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 5.75iT - 97T^{2} \) |
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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
−11.80083827812991706326798912657, −11.24005702161919798301197709897, −10.30309650447473101563607437781, −8.455346676479828741713072289058, −7.992421397171080527281841133933, −6.93852451913305315095295778209, −5.76087008000104562991491297249, −5.41486469359343751209725891821, −3.02912712124446295139523435554, −1.53904908454977666008689926132,
1.66113178040429960796949228442, 3.46916923692081123162437154598, 4.57730827895147542510672375041, 5.72478468799690319143799692644, 7.01272140186193334247575804713, 7.891059387611582640586139087103, 9.586396845248500250710920137940, 10.14786132886761486807768891398, 10.91370840515745840349271649532, 11.95120678779060162230037472919