| L(s) = 1 | + 1.19i·2-s + i·3-s + 0.570·4-s + (0.849 − 2.06i)5-s − 1.19·6-s − 3.76i·7-s + 3.07i·8-s − 9-s + (2.47 + 1.01i)10-s + 0.570i·12-s − 1.11i·13-s + 4.49·14-s + (2.06 + 0.849i)15-s − 2.53·16-s − 5.57i·17-s − 1.19i·18-s + ⋯ |
| L(s) = 1 | + 0.845i·2-s + 0.577i·3-s + 0.285·4-s + (0.379 − 0.925i)5-s − 0.488·6-s − 1.42i·7-s + 1.08i·8-s − 0.333·9-s + (0.782 + 0.321i)10-s + 0.164i·12-s − 0.308i·13-s + 1.20·14-s + (0.534 + 0.219i)15-s − 0.633·16-s − 1.35i·17-s − 0.281i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.830510265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.830510265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-0.849 + 2.06i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.19iT - 2T^{2} \) |
| 7 | \( 1 + 3.76iT - 7T^{2} \) |
| 13 | \( 1 + 1.11iT - 13T^{2} \) |
| 17 | \( 1 + 5.57iT - 17T^{2} \) |
| 19 | \( 1 + 1.37T + 19T^{2} \) |
| 23 | \( 1 + 2.35iT - 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 9.94T + 31T^{2} \) |
| 37 | \( 1 - 4.01iT - 37T^{2} \) |
| 41 | \( 1 + 3.94T + 41T^{2} \) |
| 43 | \( 1 + 7.78iT - 43T^{2} \) |
| 47 | \( 1 + 6.46iT - 47T^{2} \) |
| 53 | \( 1 - 4.06iT - 53T^{2} \) |
| 59 | \( 1 - 4.28T + 59T^{2} \) |
| 61 | \( 1 - 8.32T + 61T^{2} \) |
| 67 | \( 1 + 9.15iT - 67T^{2} \) |
| 71 | \( 1 + 2.85T + 71T^{2} \) |
| 73 | \( 1 + 11.6iT - 73T^{2} \) |
| 79 | \( 1 + 3.07T + 79T^{2} \) |
| 83 | \( 1 - 4.64iT - 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 18.1iT - 97T^{2} \) |
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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
−9.097019500414725313369881335989, −8.337067592592301422195514581079, −7.57719741723798879711765983650, −6.89780365946874663696068835872, −6.01866747264857805662048198911, −5.10153657564115247320896253595, −4.58585064088850549452676988090, −3.48999355302586391687283841821, −2.15094448357204167713223125485, −0.63616842835739201742640803861,
1.62451535187303954401403002393, 2.25998748086015675501048206095, 3.00894975312500219997915308549, 3.97553727974918941159504958357, 5.57813245757896804753236973337, 6.15674917654320528591794378489, 6.78184448588881694810972057884, 7.72669238750404504529438683057, 8.611991237758993966249945885985, 9.516782010897097510824751824696
