| L(s) = 1 | + (−0.696 − 1.58i)3-s − 2.04i·5-s − 2.04i·7-s + (−2.03 + 2.20i)9-s − 2.66·11-s − 5.30·13-s + (−3.25 + 1.42i)15-s + 6.71i·17-s − 0.878i·19-s + (−3.25 + 1.42i)21-s − 23-s + 0.799·25-s + (4.91 + 1.68i)27-s + 3.68i·29-s − 3.90i·31-s + ⋯ |
| L(s) = 1 | + (−0.401 − 0.915i)3-s − 0.916i·5-s − 0.774i·7-s + (−0.676 + 0.736i)9-s − 0.804·11-s − 1.47·13-s + (−0.839 + 0.368i)15-s + 1.62i·17-s − 0.201i·19-s + (−0.709 + 0.311i)21-s − 0.208·23-s + 0.159·25-s + (0.946 + 0.323i)27-s + 0.683i·29-s − 0.701i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.401 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1579656406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1579656406\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.696 + 1.58i)T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2.04iT - 5T^{2} \) |
| 7 | \( 1 + 2.04iT - 7T^{2} \) |
| 11 | \( 1 + 2.66T + 11T^{2} \) |
| 13 | \( 1 + 5.30T + 13T^{2} \) |
| 17 | \( 1 - 6.71iT - 17T^{2} \) |
| 19 | \( 1 + 0.878iT - 19T^{2} \) |
| 29 | \( 1 - 3.68iT - 29T^{2} \) |
| 31 | \( 1 + 3.90iT - 31T^{2} \) |
| 37 | \( 1 + 0.668T + 37T^{2} \) |
| 41 | \( 1 - 1.92iT - 41T^{2} \) |
| 43 | \( 1 + 9.63iT - 43T^{2} \) |
| 47 | \( 1 - 5.97T + 47T^{2} \) |
| 53 | \( 1 + 0.878iT - 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 + 3.33T + 61T^{2} \) |
| 67 | \( 1 - 12.5iT - 67T^{2} \) |
| 71 | \( 1 + 8.43T + 71T^{2} \) |
| 73 | \( 1 + 8.84T + 73T^{2} \) |
| 79 | \( 1 + 3.22iT - 79T^{2} \) |
| 83 | \( 1 - 9.28T + 83T^{2} \) |
| 89 | \( 1 - 13.7iT - 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−9.165610526906287481185079310583, −8.273101385416675872107474631741, −7.60746645992787521932680814785, −6.92941137118279658314487550304, −5.80552873878033134258228073710, −5.07117219466375241090415530749, −4.16356587436796879170461510936, −2.59094914047123591403788260626, −1.41993843509633246861762167060, −0.07192071616955451807227617322,
2.59626906887816559968137818464, 3.01703102529413903096297544039, 4.56243071552629935965925966870, 5.19186949466506146136443193357, 6.06803969636657458444263516749, 7.07621954726771990300612154830, 7.85186298225847907270207920336, 9.106734616676789925757921034528, 9.632529985360217367402725329484, 10.41934082227561759220003837202
