| L(s) = 1 | − 1.73i·3-s − 2.23i·5-s + (3.33 + 6.15i)7-s − 2.99·9-s − 17.0·11-s − 16.3i·13-s − 3.87·15-s − 13.4i·17-s + 13.7i·19-s + (10.6 − 5.77i)21-s + 16.6·23-s − 5.00·25-s + 5.19i·27-s + 32.1·29-s − 6.74i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.447i·5-s + (0.476 + 0.879i)7-s − 0.333·9-s − 1.54·11-s − 1.25i·13-s − 0.258·15-s − 0.789i·17-s + 0.723i·19-s + (0.507 − 0.274i)21-s + 0.722·23-s − 0.200·25-s + 0.192i·27-s + 1.10·29-s − 0.217i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.476 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3103220492\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3103220492\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + (-3.33 - 6.15i)T \) |
| good | 11 | \( 1 + 17.0T + 121T^{2} \) |
| 13 | \( 1 + 16.3iT - 169T^{2} \) |
| 17 | \( 1 + 13.4iT - 289T^{2} \) |
| 19 | \( 1 - 13.7iT - 361T^{2} \) |
| 23 | \( 1 - 16.6T + 529T^{2} \) |
| 29 | \( 1 - 32.1T + 841T^{2} \) |
| 31 | \( 1 + 6.74iT - 961T^{2} \) |
| 37 | \( 1 + 69.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 39.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 43.2T + 1.84e3T^{2} \) |
| 47 | \( 1 - 40.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 22.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 81.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 14.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 72.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 25.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + 75.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 80.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 128. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 159. iT - 9.40e3T^{2} \) |
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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
−9.292183603634946938628207436893, −8.345947569465254224770415342582, −8.045203256533589358842557903790, −7.21431097693000995404239962019, −6.04132210679192730744093161692, −5.28304988126294011147850075968, −4.86715112961529201259137052486, −3.13938724544443172830577408647, −2.50632634557836163893536624868, −1.24003720617806453485219885436,
0.082407087811333369522027848974, 1.76938824931931962026104543804, 2.91183535958073166828565078662, 3.89509090141257515464790653267, 4.76380166280726332984121437924, 5.40526568191328107960306629834, 6.73743191072829232693995061188, 7.16547214790321890075644975573, 8.255865172229951665946466916609, 8.782997569376913479236127235937
