| L(s) = 1 | − 3-s + 1.52i·5-s + i·7-s + 9-s + 1.09i·11-s + (−0.311 + 3.59i)13-s − 1.52i·15-s − 4.42·17-s + 1.80i·19-s − i·21-s + 3.80·23-s + 2.67·25-s − 27-s − 0.755·29-s + 4.85i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.682i·5-s + 0.377i·7-s + 0.333·9-s + 0.330i·11-s + (−0.0862 + 0.996i)13-s − 0.393i·15-s − 1.07·17-s + 0.414i·19-s − 0.218i·21-s + 0.793·23-s + 0.534·25-s − 0.192·27-s − 0.140·29-s + 0.872i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7961238231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7961238231\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.311 - 3.59i)T \) |
| good | 5 | \( 1 - 1.52iT - 5T^{2} \) |
| 11 | \( 1 - 1.09iT - 11T^{2} \) |
| 17 | \( 1 + 4.42T + 17T^{2} \) |
| 19 | \( 1 - 1.80iT - 19T^{2} \) |
| 23 | \( 1 - 3.80T + 23T^{2} \) |
| 29 | \( 1 + 0.755T + 29T^{2} \) |
| 31 | \( 1 - 4.85iT - 31T^{2} \) |
| 37 | \( 1 + 5.80iT - 37T^{2} \) |
| 41 | \( 1 - 11.3iT - 41T^{2} \) |
| 43 | \( 1 - 5.24T + 43T^{2} \) |
| 47 | \( 1 - 2.28iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 0.474iT - 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 9.80iT - 67T^{2} \) |
| 71 | \( 1 - 13.0iT - 71T^{2} \) |
| 73 | \( 1 - 3.47iT - 73T^{2} \) |
| 79 | \( 1 - 5.37T + 79T^{2} \) |
| 83 | \( 1 + 13.8iT - 83T^{2} \) |
| 89 | \( 1 + 13.1iT - 89T^{2} \) |
| 97 | \( 1 - 4.42iT - 97T^{2} \) |
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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
−8.942807997233955783089061563608, −7.84312782417872315063879358422, −7.08937370302547694789277378990, −6.54440038238085309836613963541, −5.97353965332294026780998655411, −4.88732676584149752009790038961, −4.42094822753679360669339917159, −3.30430087422125522506634873751, −2.40680976541065475822602636409, −1.43382027322472992792863302116,
0.26736882370630471092962832615, 1.13098740940648607438933020389, 2.42637069234976860103099102607, 3.46045356425964481059433798558, 4.44848079667537569589754866329, 5.00303160326537445137092529214, 5.73920676809049623909858096807, 6.53005135444883196308387980108, 7.25925365947874855075916101541, 7.997712765873742447564893999398
