L(s) = 1 | + (−1.28 + 0.599i)2-s − 3-s + (1.28 − 1.53i)4-s + 3.33i·5-s + (1.28 − 0.599i)6-s + (−1.56 + 2.13i)7-s + (−0.719 + 2.73i)8-s + 9-s + (−2 − 4.27i)10-s + 0.936i·11-s + (−1.28 + 1.53i)12-s + 1.87i·13-s + (0.719 − 3.67i)14-s − 3.33i·15-s + (−0.719 − 3.93i)16-s − 5.20i·17-s + ⋯ |
L(s) = 1 | + (−0.905 + 0.424i)2-s − 0.577·3-s + (0.640 − 0.768i)4-s + 1.49i·5-s + (0.522 − 0.244i)6-s + (−0.590 + 0.807i)7-s + (−0.254 + 0.967i)8-s + 0.333·9-s + (−0.632 − 1.35i)10-s + 0.282i·11-s + (−0.369 + 0.443i)12-s + 0.519i·13-s + (0.192 − 0.981i)14-s − 0.861i·15-s + (−0.179 − 0.983i)16-s − 1.26i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.315661 + 0.404080i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.315661 + 0.404080i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.28 - 0.599i)T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + (1.56 - 2.13i)T \) |
good | 5 | \( 1 - 3.33iT - 5T^{2} \) |
| 11 | \( 1 - 0.936iT - 11T^{2} \) |
| 13 | \( 1 - 1.87iT - 13T^{2} \) |
| 17 | \( 1 + 5.20iT - 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 + 0.936iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 1.12T + 37T^{2} \) |
| 41 | \( 1 + 1.46iT - 41T^{2} \) |
| 43 | \( 1 - 9.06iT - 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 4.79iT - 61T^{2} \) |
| 67 | \( 1 + 10.9iT - 67T^{2} \) |
| 71 | \( 1 + 3.86iT - 71T^{2} \) |
| 73 | \( 1 - 6.67iT - 73T^{2} \) |
| 79 | \( 1 + 2.39iT - 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 1.46iT - 89T^{2} \) |
| 97 | \( 1 + 10.4iT - 97T^{2} \) |
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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
−14.86608791623833703678971231846, −13.84321846689656638694858760498, −11.93192766094129542846043672105, −11.26048853340563633005253091148, −10.05605089236899779108555143567, −9.284772975050576703196082624885, −7.43224542851742676728711725785, −6.69264289318017612243495303152, −5.53586871994434092366007764959, −2.74612645231323554418663994176,
0.985351524338355910375931631778, 3.85272822630576510105703056642, 5.65037187613062237570453708006, 7.30083954693586896562771172570, 8.485088658869101834459097857355, 9.615539218424547144561418829750, 10.54090715318362925293792215804, 11.80808477627811878131244733606, 12.70023491209255873490481778914, 13.45024545278687333097783788502