L(s) = 1 | + i·2-s − 4-s − 1.41i·5-s + (2.12 − 1.58i)7-s − i·8-s + 1.41·10-s + (3.16 − i)11-s + 1.41·13-s + (1.58 + 2.12i)14-s + 16-s − 4.47·17-s − 2.82·19-s + 1.41i·20-s + (1 + 3.16i)22-s − 3.16·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.632i·5-s + (0.801 − 0.597i)7-s − 0.353i·8-s + 0.447·10-s + (0.953 − 0.301i)11-s + 0.392·13-s + (0.422 + 0.566i)14-s + 0.250·16-s − 1.08·17-s − 0.648·19-s + 0.316i·20-s + (0.213 + 0.674i)22-s − 0.659·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.603405956\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.603405956\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.12 + 1.58i)T \) |
| 11 | \( 1 + (-3.16 + i)T \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 8.94iT - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 8.94T + 41T^{2} \) |
| 43 | \( 1 - 6.32iT - 43T^{2} \) |
| 47 | \( 1 + 7.07iT - 47T^{2} \) |
| 53 | \( 1 - 3.16T + 53T^{2} \) |
| 59 | \( 1 + 2.82iT - 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 3.16T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 3.16iT - 79T^{2} \) |
| 83 | \( 1 + 4.47T + 83T^{2} \) |
| 89 | \( 1 - 16.9iT - 89T^{2} \) |
| 97 | \( 1 + 4.47iT - 97T^{2} \) |
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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
−9.313697382136220073941564692161, −8.396507538244807043666439603113, −8.152667491648556076129406771545, −6.93522782627502510616488757148, −6.33312520057005806417420313190, −5.33039901824254557769240847800, −4.34577064510933633718103796707, −3.91714453025118099529218687995, −2.03079318828780466977323046928, −0.69875863011358737057608572246,
1.48050130065285066296025670521, 2.38298745578670658495675962591, 3.51382955987763344274131902206, 4.44775794299829619640025740382, 5.31327047021692480240544431032, 6.48384298036468676835204001755, 7.08127276121211981007450566265, 8.554897087825962010083055997909, 8.653389063101979321030231653125, 9.725048298170614440293215811319