L(s) = 1 | + (−0.565 − 1.29i)2-s + (−1.40 − 1.01i)3-s + (−1.35 + 1.46i)4-s − 0.662i·5-s + (−0.524 + 2.39i)6-s + 4.60i·7-s + (2.67 + 0.933i)8-s + (0.932 + 2.85i)9-s + (−0.859 + 0.375i)10-s − 1.50·11-s + (3.39 − 0.673i)12-s + 1.18·13-s + (5.96 − 2.60i)14-s + (−0.673 + 0.929i)15-s + (−0.301 − 3.98i)16-s + 1.95i·17-s + ⋯ |
L(s) = 1 | + (−0.400 − 0.916i)2-s + (−0.809 − 0.586i)3-s + (−0.679 + 0.733i)4-s − 0.296i·5-s + (−0.214 + 0.976i)6-s + 1.73i·7-s + (0.944 + 0.329i)8-s + (0.310 + 0.950i)9-s + (−0.271 + 0.118i)10-s − 0.452·11-s + (0.980 − 0.194i)12-s + 0.328·13-s + (1.59 − 0.695i)14-s + (−0.174 + 0.240i)15-s + (−0.0752 − 0.997i)16-s + 0.473i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.624170 + 0.0612981i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.624170 + 0.0612981i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.565 + 1.29i)T \) |
| 3 | \( 1 + (1.40 + 1.01i)T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 0.662iT - 5T^{2} \) |
| 7 | \( 1 - 4.60iT - 7T^{2} \) |
| 11 | \( 1 + 1.50T + 11T^{2} \) |
| 13 | \( 1 - 1.18T + 13T^{2} \) |
| 17 | \( 1 - 1.95iT - 17T^{2} \) |
| 19 | \( 1 - 0.0109iT - 19T^{2} \) |
| 29 | \( 1 - 8.65iT - 29T^{2} \) |
| 31 | \( 1 - 6.06iT - 31T^{2} \) |
| 37 | \( 1 + 7.04T + 37T^{2} \) |
| 41 | \( 1 - 7.37iT - 41T^{2} \) |
| 43 | \( 1 + 4.67iT - 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 0.825iT - 53T^{2} \) |
| 59 | \( 1 + 6.63T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 - 0.812iT - 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 - 9.43iT - 79T^{2} \) |
| 83 | \( 1 + 1.34T + 83T^{2} \) |
| 89 | \( 1 + 6.30iT - 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−12.04226820272546890475459364349, −11.06816837097714288113866201495, −10.29787754597035696382576845068, −8.884712505476807611277427007830, −8.471186634504484739337800723279, −7.07007461749824857035994354645, −5.66164403284029512049847572352, −4.88447455402747770673177824454, −2.93375596146770948873597098815, −1.60936666426185048087575512538,
0.64685822399562259874930906415, 3.86751924330338071703542910833, 4.75911786388651231786771409820, 5.97364770672363756687478864293, 6.97017919236724815909508489838, 7.66405066012525304105823236513, 9.074193984450898167920989164456, 10.19820543206485972660769944713, 10.52427346927226579375725764708, 11.49456679605829695781496431936