L(s) = 1 | + (−1.29 − 0.578i)2-s + (1.56 + 0.751i)3-s + (1.33 + 1.49i)4-s + (−1.57 − 1.87i)6-s + 4.28·7-s + (−0.852 − 2.69i)8-s + (1.86 + 2.34i)9-s + 2.44i·11-s + (0.953 + 3.33i)12-s + 2.71·13-s + (−5.53 − 2.48i)14-s + (−0.460 + 3.97i)16-s − 1.16·17-s + (−1.05 − 4.10i)18-s − 6.05·19-s + ⋯ |
L(s) = 1 | + (−0.912 − 0.409i)2-s + (0.900 + 0.433i)3-s + (0.665 + 0.746i)4-s + (−0.644 − 0.764i)6-s + 1.61·7-s + (−0.301 − 0.953i)8-s + (0.623 + 0.781i)9-s + 0.737i·11-s + (0.275 + 0.961i)12-s + 0.752·13-s + (−1.47 − 0.662i)14-s + (−0.115 + 0.993i)16-s − 0.282·17-s + (−0.248 − 0.968i)18-s − 1.38·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.55866 + 0.252267i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55866 + 0.252267i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.29 + 0.578i)T \) |
| 3 | \( 1 + (-1.56 - 0.751i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4.28T + 7T^{2} \) |
| 11 | \( 1 - 2.44iT - 11T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 + 1.16T + 17T^{2} \) |
| 19 | \( 1 + 6.05T + 19T^{2} \) |
| 23 | \( 1 + 7.55iT - 23T^{2} \) |
| 29 | \( 1 + 0.733T + 29T^{2} \) |
| 31 | \( 1 - 0.469iT - 31T^{2} \) |
| 37 | \( 1 + 1.36T + 37T^{2} \) |
| 41 | \( 1 - 4.69iT - 41T^{2} \) |
| 43 | \( 1 + 1.50iT - 43T^{2} \) |
| 47 | \( 1 - 4.07iT - 47T^{2} \) |
| 53 | \( 1 - 1.00iT - 53T^{2} \) |
| 59 | \( 1 - 1.63iT - 59T^{2} \) |
| 61 | \( 1 + 10.9iT - 61T^{2} \) |
| 67 | \( 1 - 9.97iT - 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 9.63iT - 73T^{2} \) |
| 79 | \( 1 - 3.61iT - 79T^{2} \) |
| 83 | \( 1 - 5.45T + 83T^{2} \) |
| 89 | \( 1 + 7.75iT - 89T^{2} \) |
| 97 | \( 1 + 17.1iT - 97T^{2} \) |
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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
−10.76226107748931575335712112733, −9.824052922796031793272867979065, −8.712966052594638713179886916147, −8.403096035400835585165892083891, −7.60884542568331911707002863606, −6.53074545927234306808063975894, −4.74638825656716696029849151310, −4.02256594766040756435974880411, −2.48335847960520698040051565479, −1.64447049002211886703075764421,
1.29474823675554038755688075771, 2.23812153398950446355758649975, 3.85529984253917858133448275442, 5.29029019941078811071171501998, 6.36730431583782696354877568799, 7.38950137176100375981051480448, 8.199942684919080528741396444867, 8.574382221841158625514294300756, 9.404546912145187868627893902107, 10.66078972937590878162150930667