L(s) = 1 | + (−1.22 − 1.22i)3-s + (−3.67 + 3.67i)7-s + 2.99i·9-s − 6·11-s + (6.12 + 6.12i)13-s + (22.0 − 22.0i)17-s − 25i·19-s + 9·21-s + (7.34 + 7.34i)23-s + (3.67 − 3.67i)27-s + 42i·29-s − 49·31-s + (7.34 + 7.34i)33-s + (4.89 − 4.89i)37-s − 14.9i·39-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (−0.524 + 0.524i)7-s + 0.333i·9-s − 0.545·11-s + (0.471 + 0.471i)13-s + (1.29 − 1.29i)17-s − 1.31i·19-s + 0.428·21-s + (0.319 + 0.319i)23-s + (0.136 − 0.136i)27-s + 1.44i·29-s − 1.58·31-s + (0.222 + 0.222i)33-s + (0.132 − 0.132i)37-s − 0.384i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6788203488\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6788203488\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (3.67 - 3.67i)T - 49iT^{2} \) |
| 11 | \( 1 + 6T + 121T^{2} \) |
| 13 | \( 1 + (-6.12 - 6.12i)T + 169iT^{2} \) |
| 17 | \( 1 + (-22.0 + 22.0i)T - 289iT^{2} \) |
| 19 | \( 1 + 25iT - 361T^{2} \) |
| 23 | \( 1 + (-7.34 - 7.34i)T + 529iT^{2} \) |
| 29 | \( 1 - 42iT - 841T^{2} \) |
| 31 | \( 1 + 49T + 961T^{2} \) |
| 37 | \( 1 + (-4.89 + 4.89i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 60T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-1.22 - 1.22i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (51.4 - 51.4i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (14.6 + 14.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 78iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 13T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-52.6 + 52.6i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 60T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-63.6 - 63.6i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 106iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-80.8 - 80.8i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 60iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (121. - 121. i)T - 9.40e3iT^{2} \) |
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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
−9.591141170524219895373420033383, −9.170917759442954622463352966489, −8.099525267310405960744139596492, −7.16860336998673250049256423030, −6.62151058570814266215166251512, −5.43084708407577714365117419896, −5.04047541695511024727871454620, −3.46994776856508752309053620140, −2.60820295195488558602547876954, −1.19519753946113291602599968571,
0.23028244290059556018683163379, 1.69142531376407168140476160628, 3.41026988846220617941485397927, 3.82355370038079837736667357781, 5.18806233982789407191048920131, 5.87113081221152322661267972712, 6.66309176233734348364999375297, 7.84727067127762692901748973598, 8.310800671997849032489396968245, 9.600987579577636772125546402006