| L(s) = 1 | − 3-s − 3i·5-s + i·7-s + 9-s − 4i·11-s + (2 + 3i)13-s + 3i·15-s + 6·17-s + 7i·19-s − i·21-s − 23-s − 4·25-s − 27-s + 29-s + 7i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34i·5-s + 0.377i·7-s + 0.333·9-s − 1.20i·11-s + (0.554 + 0.832i)13-s + 0.774i·15-s + 1.45·17-s + 1.60i·19-s − 0.218i·21-s − 0.208·23-s − 0.800·25-s − 0.192·27-s + 0.185·29-s + 1.25i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.384965659\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.384965659\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 + (-2 - 3i)T \) |
| good | 5 | \( 1 + 3iT - 5T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 7iT - 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - T + 29T^{2} \) |
| 31 | \( 1 - 7iT - 31T^{2} \) |
| 37 | \( 1 - 12iT - 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 + 11T + 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 6iT - 67T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 - 11iT - 73T^{2} \) |
| 79 | \( 1 - 15T + 79T^{2} \) |
| 83 | \( 1 - 15iT - 83T^{2} \) |
| 89 | \( 1 + 11iT - 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−8.382727049596025346340562136311, −8.045705607407691659436078627894, −6.81892355778058308698077354679, −6.03633762594360331831201518923, −5.48660355724880677981472902540, −4.91921661782693105269749600413, −3.90650036721451309976714546461, −3.22335136541662494513164020154, −1.56888561835828550203111598133, −1.06947932463898115256300323382,
0.49330282133277651099906127829, 1.88597987254858056082741587211, 2.92446947994951148400058396319, 3.61505582217247546456430387106, 4.61737260061205569487576201501, 5.37658986480747634154607529367, 6.28699067582588991221216093683, 6.76657056581834794571008738836, 7.61416510260865972408388444784, 7.84537150559380198327466438679
