L(s) = 1 | + (1.39 + 0.221i)2-s + 2.04i·3-s + (1.90 + 0.619i)4-s + (0.665 − 2.13i)5-s + (−0.453 + 2.85i)6-s + 2.07·7-s + (2.51 + 1.28i)8-s − 1.17·9-s + (1.40 − 2.83i)10-s − 4.08i·11-s + (−1.26 + 3.88i)12-s − 4.52·13-s + (2.90 + 0.460i)14-s + (4.36 + 1.36i)15-s + (3.23 + 2.35i)16-s + 6.58i·17-s + ⋯ |
L(s) = 1 | + (0.987 + 0.156i)2-s + 1.17i·3-s + (0.950 + 0.309i)4-s + (0.297 − 0.954i)5-s + (−0.184 + 1.16i)6-s + 0.785·7-s + (0.890 + 0.454i)8-s − 0.392·9-s + (0.443 − 0.896i)10-s − 1.23i·11-s + (−0.365 + 1.12i)12-s − 1.25·13-s + (0.775 + 0.123i)14-s + (1.12 + 0.351i)15-s + (0.808 + 0.588i)16-s + 1.59i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.42026 + 1.04875i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.42026 + 1.04875i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.39 - 0.221i)T \) |
| 5 | \( 1 + (-0.665 + 2.13i)T \) |
| 19 | \( 1 + (4.31 + 0.590i)T \) |
good | 3 | \( 1 - 2.04iT - 3T^{2} \) |
| 7 | \( 1 - 2.07T + 7T^{2} \) |
| 11 | \( 1 + 4.08iT - 11T^{2} \) |
| 13 | \( 1 + 4.52T + 13T^{2} \) |
| 17 | \( 1 - 6.58iT - 17T^{2} \) |
| 23 | \( 1 + 7.09T + 23T^{2} \) |
| 29 | \( 1 + 5.90iT - 29T^{2} \) |
| 31 | \( 1 - 1.69T + 31T^{2} \) |
| 37 | \( 1 - 1.81T + 37T^{2} \) |
| 41 | \( 1 - 1.10iT - 41T^{2} \) |
| 43 | \( 1 + 7.73T + 43T^{2} \) |
| 47 | \( 1 - 9.98T + 47T^{2} \) |
| 53 | \( 1 - 2.18T + 53T^{2} \) |
| 59 | \( 1 - 9.39T + 59T^{2} \) |
| 61 | \( 1 - 4.95T + 61T^{2} \) |
| 67 | \( 1 + 13.3iT - 67T^{2} \) |
| 71 | \( 1 + 4.67T + 71T^{2} \) |
| 73 | \( 1 - 6.18iT - 73T^{2} \) |
| 79 | \( 1 - 4.04T + 79T^{2} \) |
| 83 | \( 1 - 7.46T + 83T^{2} \) |
| 89 | \( 1 - 0.553iT - 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−11.55028454140791561192811260112, −10.59038654330129441219197769904, −9.894995806925035194518229606823, −8.539305006683586120412756587322, −7.966614089533617144183464913843, −6.22286214043477290735230197081, −5.40224315860310932675060719385, −4.48360836027624414688477889259, −3.86849422883725166768225577746, −2.08165822484705220983609965364,
1.92386545280533936808911526355, 2.53328981351045027220246295423, 4.33753441497844683010545610516, 5.37858624387610118693707392943, 6.70735415778507551017322514862, 7.15191565557267125321277693919, 7.86616476600852292331955833634, 9.770352362317835469345771882595, 10.47377301367104685316128680677, 11.74519190513506721917362050313