| L(s) = 1 | + i·3-s + i·5-s + 1.58·7-s − 9-s − 2.94·11-s − 1.23·13-s − 15-s + 1.25i·17-s + 5.26·19-s + 1.58i·21-s + (1.83 + 4.42i)23-s − 25-s − i·27-s + 7.54·29-s + 0.690i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.447i·5-s + 0.598·7-s − 0.333·9-s − 0.887·11-s − 0.341·13-s − 0.258·15-s + 0.304i·17-s + 1.20·19-s + 0.345i·21-s + (0.383 + 0.923i)23-s − 0.200·25-s − 0.192i·27-s + 1.40·29-s + 0.123i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.751304679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.751304679\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-1.83 - 4.42i)T \) |
| good | 7 | \( 1 - 1.58T + 7T^{2} \) |
| 11 | \( 1 + 2.94T + 11T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 1.25iT - 17T^{2} \) |
| 19 | \( 1 - 5.26T + 19T^{2} \) |
| 29 | \( 1 - 7.54T + 29T^{2} \) |
| 31 | \( 1 - 0.690iT - 31T^{2} \) |
| 37 | \( 1 + 4.83iT - 37T^{2} \) |
| 41 | \( 1 - 6.98T + 41T^{2} \) |
| 43 | \( 1 - 4.06T + 43T^{2} \) |
| 47 | \( 1 + 8.95iT - 47T^{2} \) |
| 53 | \( 1 - 10.8iT - 53T^{2} \) |
| 59 | \( 1 + 2.33iT - 59T^{2} \) |
| 61 | \( 1 - 5.28iT - 61T^{2} \) |
| 67 | \( 1 + 9.49T + 67T^{2} \) |
| 71 | \( 1 - 13.5iT - 71T^{2} \) |
| 73 | \( 1 + 9.51T + 73T^{2} \) |
| 79 | \( 1 + 3.71T + 79T^{2} \) |
| 83 | \( 1 - 16.5T + 83T^{2} \) |
| 89 | \( 1 - 12.2iT - 89T^{2} \) |
| 97 | \( 1 - 7.70iT - 97T^{2} \) |
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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
−8.323852285696986702581141115602, −7.61234962412136094556003623671, −7.17870751819792624808440942383, −6.06088565813359197351503864999, −5.40349074811235695587908164113, −4.81597320647811743837852491572, −3.95770167642836362421346641731, −3.05063389705336210894068563548, −2.39917431541748403474214972204, −1.09814245840356466900920876992,
0.51363191093722892102691548385, 1.46242632379710099817267222908, 2.54469153203757754167278115665, 3.16624791961255659125918255991, 4.61908970772629032337442000902, 4.87195568691484583272371645259, 5.77483597097659726830735286852, 6.51516576557191316803428257813, 7.43987583711666522063237663444, 7.84212133488670455763277138925
