L(s) = 1 | + (−0.619 − 1.27i)2-s + (−1.23 + 1.57i)4-s − 0.753·5-s − i·7-s + (2.76 + 0.590i)8-s + (0.467 + 0.958i)10-s − 4.40i·11-s + 4.28i·13-s + (−1.27 + 0.619i)14-s + (−0.963 − 3.88i)16-s − 7.48i·17-s + 0.157·19-s + (0.928 − 1.18i)20-s + (−5.59 + 2.72i)22-s − 2.84·23-s + ⋯ |
L(s) = 1 | + (−0.438 − 0.898i)2-s + (−0.616 + 0.787i)4-s − 0.337·5-s − 0.377i·7-s + (0.977 + 0.208i)8-s + (0.147 + 0.302i)10-s − 1.32i·11-s + 1.18i·13-s + (−0.339 + 0.165i)14-s + (−0.240 − 0.970i)16-s − 1.81i·17-s + 0.0360·19-s + (0.207 − 0.265i)20-s + (−1.19 + 0.581i)22-s − 0.592·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.139844 - 0.680574i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.139844 - 0.680574i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.619 + 1.27i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 0.753T + 5T^{2} \) |
| 11 | \( 1 + 4.40iT - 11T^{2} \) |
| 13 | \( 1 - 4.28iT - 13T^{2} \) |
| 17 | \( 1 + 7.48iT - 17T^{2} \) |
| 19 | \( 1 - 0.157T + 19T^{2} \) |
| 23 | \( 1 + 2.84T + 23T^{2} \) |
| 29 | \( 1 + 5.03T + 29T^{2} \) |
| 31 | \( 1 + 9.13iT - 31T^{2} \) |
| 37 | \( 1 + 6.38iT - 37T^{2} \) |
| 41 | \( 1 + 2.93iT - 41T^{2} \) |
| 43 | \( 1 - 1.50T + 43T^{2} \) |
| 47 | \( 1 - 6.17T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 6.17iT - 59T^{2} \) |
| 61 | \( 1 - 9.34iT - 61T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 - 2.69T + 71T^{2} \) |
| 73 | \( 1 - 1.74T + 73T^{2} \) |
| 79 | \( 1 + 12.2iT - 79T^{2} \) |
| 83 | \( 1 + 8.29iT - 83T^{2} \) |
| 89 | \( 1 - 5.13iT - 89T^{2} \) |
| 97 | \( 1 - 5.83T + 97T^{2} \) |
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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
−10.69707985823576297541467706655, −9.513799111351910200011357608863, −9.076985537783183921054287891609, −7.903086997404616246073480871749, −7.22156448191883709070314478798, −5.77375608602258764343375992090, −4.38825268212074545855561432512, −3.56543931050632047677628514349, −2.25157468598940211757275104033, −0.48331166068812345472946623691,
1.76411412004303460547606762557, 3.73968211418310691983399841450, 4.90583847394755695733378879356, 5.85804420009155458877033339905, 6.79493334513059985293748936941, 7.902030420275174867928383982676, 8.286756741759320165645894970259, 9.532491616088246555345429481433, 10.18471809820480667182390715141, 11.02604078781168144450338198625