| L(s) = 1 | − i·2-s − 4-s + 1.63i·5-s + (1.91 + 1.82i)7-s + i·8-s + 1.63·10-s + 2.23·11-s − 0.801·13-s + (1.82 − 1.91i)14-s + 16-s − 2.71i·17-s + (−3.13 − 3.02i)19-s − 1.63i·20-s − 2.23i·22-s + 6.49·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 0.733i·5-s + (0.722 + 0.691i)7-s + 0.353i·8-s + 0.518·10-s + 0.672·11-s − 0.222·13-s + (0.488 − 0.511i)14-s + 0.250·16-s − 0.658i·17-s + (−0.719 − 0.694i)19-s − 0.366i·20-s − 0.475i·22-s + 1.35·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.897598319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.897598319\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.91 - 1.82i)T \) |
| 19 | \( 1 + (3.13 + 3.02i)T \) |
| good | 5 | \( 1 - 1.63iT - 5T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 0.801T + 13T^{2} \) |
| 17 | \( 1 + 2.71iT - 17T^{2} \) |
| 23 | \( 1 - 6.49T + 23T^{2} \) |
| 29 | \( 1 + 3.82iT - 29T^{2} \) |
| 31 | \( 1 + 5.19T + 31T^{2} \) |
| 37 | \( 1 - 7.19iT - 37T^{2} \) |
| 41 | \( 1 - 6.99T + 41T^{2} \) |
| 43 | \( 1 - 7.13T + 43T^{2} \) |
| 47 | \( 1 - 1.91iT - 47T^{2} \) |
| 53 | \( 1 - 1.19iT - 53T^{2} \) |
| 59 | \( 1 - 4.59T + 59T^{2} \) |
| 61 | \( 1 - 0.934iT - 61T^{2} \) |
| 67 | \( 1 - 15.2iT - 67T^{2} \) |
| 71 | \( 1 - 1.19iT - 71T^{2} \) |
| 73 | \( 1 - 6.99iT - 73T^{2} \) |
| 79 | \( 1 - 0.503iT - 79T^{2} \) |
| 83 | \( 1 - 11.1iT - 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−9.055797988678626270122414671875, −8.442257463569396769382139968462, −7.39411222931616482645881127773, −6.74107217695598942205697450811, −5.73060120966821768241514275472, −4.87220105215265528832329376746, −4.11056355301484644798146987488, −2.89387507619629006767589991812, −2.38427770079802306333555473549, −1.07847807522177772978079447431,
0.809954292861860703411789714090, 1.85027646320313048413921972441, 3.55586337326353285855249073513, 4.31687883362458492482446319046, 5.00046492297733627119309885139, 5.81203561106869896787161397926, 6.74009341134090365207254570315, 7.45486850604653972121033735192, 8.126074998002885678726030543724, 8.985587869952850730017118600645
