| L(s) = 1 | + 2.45·3-s + 2.23i·5-s + 0.852i·7-s − 2.98·9-s − 0.213i·11-s + 3.44·13-s + 5.48i·15-s − 25.2i·17-s + 2.50i·19-s + 2.09i·21-s + (−15.5 − 16.9i)23-s − 5.00·25-s − 29.3·27-s − 46.4·29-s − 22.0·31-s + ⋯ |
| L(s) = 1 | + 0.817·3-s + 0.447i·5-s + 0.121i·7-s − 0.332·9-s − 0.0193i·11-s + 0.265·13-s + 0.365i·15-s − 1.48i·17-s + 0.131i·19-s + 0.0995i·21-s + (−0.675 − 0.736i)23-s − 0.200·25-s − 1.08·27-s − 1.60·29-s − 0.711·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.675i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.736 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7161924207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7161924207\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (15.5 + 16.9i)T \) |
| good | 3 | \( 1 - 2.45T + 9T^{2} \) |
| 7 | \( 1 - 0.852iT - 49T^{2} \) |
| 11 | \( 1 + 0.213iT - 121T^{2} \) |
| 13 | \( 1 - 3.44T + 169T^{2} \) |
| 17 | \( 1 + 25.2iT - 289T^{2} \) |
| 19 | \( 1 - 2.50iT - 361T^{2} \) |
| 29 | \( 1 + 46.4T + 841T^{2} \) |
| 31 | \( 1 + 22.0T + 961T^{2} \) |
| 37 | \( 1 - 45.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 6.35T + 1.68e3T^{2} \) |
| 43 | \( 1 - 6.26iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 47.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 52.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 114.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 109. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 105. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 117.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 1.67T + 5.32e3T^{2} \) |
| 79 | \( 1 + 118. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 50.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 84.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 71.4iT - 9.40e3T^{2} \) |
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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
−9.019215933018678068962461680872, −7.83778975069677774281392610497, −7.47710036500546938925600856272, −6.37328013295677221333598057887, −5.61762671984530530381325814397, −4.54932773126647527424714722231, −3.47731979580133668600484074935, −2.80088338239308746677539912488, −1.86737402726056685376491396951, −0.14997080041047814781660490583,
1.50146822618875586226781581348, 2.40078600699259259870165034818, 3.68306241373950532826723366573, 4.06419767552605337051952528775, 5.56513868763720019533815957391, 5.92106208594799873585621370073, 7.31367960305595540764228656282, 7.82321526887416593360181505700, 8.784916587763003921086235948957, 9.033248493485646691270339666528
