| L(s) = 1 | − 2.62·2-s − 1.76·3-s + 4.90·4-s + 2.94·5-s + 4.64·6-s − 7-s − 7.63·8-s + 0.124·9-s − 7.73·10-s − 4.28·11-s − 8.67·12-s + 2.62·14-s − 5.20·15-s + 10.2·16-s + 2.94·17-s − 0.327·18-s − 4.98·19-s + 14.4·20-s + 1.76·21-s + 11.2·22-s + 7.58·23-s + 13.5·24-s + 3.66·25-s + 5.08·27-s − 4.90·28-s − 1.82·29-s + 13.6·30-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 1.02·3-s + 2.45·4-s + 1.31·5-s + 1.89·6-s − 0.377·7-s − 2.70·8-s + 0.0414·9-s − 2.44·10-s − 1.29·11-s − 2.50·12-s + 0.702·14-s − 1.34·15-s + 2.56·16-s + 0.713·17-s − 0.0770·18-s − 1.14·19-s + 3.22·20-s + 0.385·21-s + 2.39·22-s + 1.58·23-s + 2.75·24-s + 0.732·25-s + 0.978·27-s − 0.927·28-s − 0.339·29-s + 2.49·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4376715048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4376715048\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 + 1.76T + 3T^{2} \) |
| 5 | \( 1 - 2.94T + 5T^{2} \) |
| 11 | \( 1 + 4.28T + 11T^{2} \) |
| 17 | \( 1 - 2.94T + 17T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 23 | \( 1 - 7.58T + 23T^{2} \) |
| 29 | \( 1 + 1.82T + 29T^{2} \) |
| 31 | \( 1 + 6.57T + 31T^{2} \) |
| 37 | \( 1 - 8.31T + 37T^{2} \) |
| 41 | \( 1 - 4.15T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 0.910T + 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 - 9.42T + 59T^{2} \) |
| 61 | \( 1 + 1.75T + 61T^{2} \) |
| 67 | \( 1 + 0.413T + 67T^{2} \) |
| 71 | \( 1 - 2.95T + 71T^{2} \) |
| 73 | \( 1 - 0.885T + 73T^{2} \) |
| 79 | \( 1 - 10.0T + 79T^{2} \) |
| 83 | \( 1 + 6.04T + 83T^{2} \) |
| 89 | \( 1 - 3.82T + 89T^{2} \) |
| 97 | \( 1 - 2.37T + 97T^{2} \) |
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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
−9.810318402660814775186590437941, −9.125825276685808169206103321379, −8.314452678956761956736496665963, −7.33908222261975723232180560985, −6.52522679380175620907133130672, −5.84037123630612587559727244087, −5.17586521516046375088840859485, −2.91616931946918808481164130363, −2.01995271220235700047968486757, −0.66433205624370383811223593158,
0.66433205624370383811223593158, 2.01995271220235700047968486757, 2.91616931946918808481164130363, 5.17586521516046375088840859485, 5.84037123630612587559727244087, 6.52522679380175620907133130672, 7.33908222261975723232180560985, 8.314452678956761956736496665963, 9.125825276685808169206103321379, 9.810318402660814775186590437941
