L(s) = 1 | + (1.28 + 0.599i)2-s + (1.28 + 1.53i)4-s + 3.33i·5-s + (−1.56 − 2.13i)7-s + (0.719 + 2.73i)8-s + (−2 + 4.27i)10-s + 0.936i·11-s − 1.87i·13-s + (−0.719 − 3.67i)14-s + (−0.719 + 3.93i)16-s − 5.20i·17-s + 7.12·19-s + (−5.12 + 4.27i)20-s + (−0.561 + 1.19i)22-s − 0.936i·23-s + ⋯ |
L(s) = 1 | + (0.905 + 0.424i)2-s + (0.640 + 0.768i)4-s + 1.49i·5-s + (−0.590 − 0.807i)7-s + (0.254 + 0.967i)8-s + (−0.632 + 1.35i)10-s + 0.282i·11-s − 0.519i·13-s + (−0.192 − 0.981i)14-s + (−0.179 + 0.983i)16-s − 1.26i·17-s + 1.63·19-s + (−1.14 + 0.955i)20-s + (−0.119 + 0.255i)22-s − 0.195i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56102 + 1.21945i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56102 + 1.21945i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 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
−12.28878083593236828633120478789, −11.36509557378059283460201858608, −10.53829538791489777696619985208, −9.587717440272898618084327129657, −7.72247501793234012286361661719, −7.14157498086366932121834693991, −6.36679677673830833547382412002, −5.05811489511074808848571505015, −3.54650692368533166345663294916, −2.81531726953070098416216895727,
1.48609842293146197234493075877, 3.23509725083278279780199678671, 4.56737414060814760194025554038, 5.50455219882482109082299537435, 6.37462802900158648085082059336, 8.013416670888288132814655353767, 9.175689151186739969366724518355, 9.826449569499366140697451345222, 11.27755472497561347083500254404, 12.09449843865262057677109861540