L(s) = 1 | + (1.37 + 0.339i)2-s + (0.567 + 1.63i)3-s + (1.76 + 0.931i)4-s + (−0.132 − 0.132i)5-s + (0.223 + 2.43i)6-s − 7-s + (2.11 + 1.87i)8-s + (−2.35 + 1.85i)9-s + (−0.137 − 0.227i)10-s + (−0.715 + 0.715i)11-s + (−0.520 + 3.42i)12-s + (0.206 + 0.206i)13-s + (−1.37 − 0.339i)14-s + (0.142 − 0.292i)15-s + (2.26 + 3.29i)16-s − 6.91i·17-s + ⋯ |
L(s) = 1 | + (0.970 + 0.239i)2-s + (0.327 + 0.944i)3-s + (0.884 + 0.465i)4-s + (−0.0594 − 0.0594i)5-s + (0.0913 + 0.995i)6-s − 0.377·7-s + (0.747 + 0.664i)8-s + (−0.785 + 0.618i)9-s + (−0.0434 − 0.0719i)10-s + (−0.215 + 0.215i)11-s + (−0.150 + 0.988i)12-s + (0.0572 + 0.0572i)13-s + (−0.366 − 0.0906i)14-s + (0.0366 − 0.0755i)15-s + (0.566 + 0.824i)16-s − 1.67i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.92294 + 1.50537i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.92294 + 1.50537i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.37 - 0.339i)T \) |
| 3 | \( 1 + (-0.567 - 1.63i)T \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (0.132 + 0.132i)T + 5iT^{2} \) |
| 11 | \( 1 + (0.715 - 0.715i)T - 11iT^{2} \) |
| 13 | \( 1 + (-0.206 - 0.206i)T + 13iT^{2} \) |
| 17 | \( 1 + 6.91iT - 17T^{2} \) |
| 19 | \( 1 + (-5.87 + 5.87i)T - 19iT^{2} \) |
| 23 | \( 1 - 6.42iT - 23T^{2} \) |
| 29 | \( 1 + (5.00 - 5.00i)T - 29iT^{2} \) |
| 31 | \( 1 - 2.52iT - 31T^{2} \) |
| 37 | \( 1 + (-6.15 + 6.15i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.19T + 41T^{2} \) |
| 43 | \( 1 + (-1.36 - 1.36i)T + 43iT^{2} \) |
| 47 | \( 1 - 0.603T + 47T^{2} \) |
| 53 | \( 1 + (6.19 + 6.19i)T + 53iT^{2} \) |
| 59 | \( 1 + (-5.00 + 5.00i)T - 59iT^{2} \) |
| 61 | \( 1 + (6.24 + 6.24i)T + 61iT^{2} \) |
| 67 | \( 1 + (-2.55 + 2.55i)T - 67iT^{2} \) |
| 71 | \( 1 - 0.808iT - 71T^{2} \) |
| 73 | \( 1 - 11.5iT - 73T^{2} \) |
| 79 | \( 1 - 3.48iT - 79T^{2} \) |
| 83 | \( 1 + (0.641 + 0.641i)T + 83iT^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 + 5.56T + 97T^{2} \) |
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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
−11.58065551968504193435725136157, −11.15205632410388430007816001798, −9.821776572866760716074270938569, −9.134652189441459020658392058636, −7.75470765014257995925673567654, −6.93069353386295272379851622635, −5.42815098094834393441673372361, −4.85837394992811531329589693989, −3.55224147167173277116552071562, −2.66490071515306391045447977717,
1.55229321619586429926983447783, 2.96279026914312990640182646885, 3.96794498027921463035968139724, 5.69745219502494195331522225886, 6.25700719480175677390036747699, 7.44200106889972237948177835029, 8.227571034065629896619979479282, 9.659934163137879286501192138423, 10.67203619670990952932214054399, 11.69320319420638803772923163639