| L(s) = 1 | − 2-s + 0.904·3-s + 4-s − 4.23·5-s − 0.904·6-s + 1.08·7-s − 8-s − 2.18·9-s + 4.23·10-s + 5.59·11-s + 0.904·12-s + 3.70·13-s − 1.08·14-s − 3.82·15-s + 16-s − 6.11·17-s + 2.18·18-s − 19-s − 4.23·20-s + 0.979·21-s − 5.59·22-s + 7.81·23-s − 0.904·24-s + 12.8·25-s − 3.70·26-s − 4.68·27-s + 1.08·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.522·3-s + 0.5·4-s − 1.89·5-s − 0.369·6-s + 0.409·7-s − 0.353·8-s − 0.727·9-s + 1.33·10-s + 1.68·11-s + 0.261·12-s + 1.02·13-s − 0.289·14-s − 0.988·15-s + 0.250·16-s − 1.48·17-s + 0.514·18-s − 0.229·19-s − 0.945·20-s + 0.213·21-s − 1.19·22-s + 1.62·23-s − 0.184·24-s + 2.57·25-s − 0.725·26-s − 0.902·27-s + 0.204·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.111685894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.111685894\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 0.904T + 3T^{2} \) |
| 5 | \( 1 + 4.23T + 5T^{2} \) |
| 7 | \( 1 - 1.08T + 7T^{2} \) |
| 11 | \( 1 - 5.59T + 11T^{2} \) |
| 13 | \( 1 - 3.70T + 13T^{2} \) |
| 17 | \( 1 + 6.11T + 17T^{2} \) |
| 23 | \( 1 - 7.81T + 23T^{2} \) |
| 29 | \( 1 + 6.05T + 29T^{2} \) |
| 31 | \( 1 - 0.604T + 31T^{2} \) |
| 37 | \( 1 + 3.09T + 37T^{2} \) |
| 41 | \( 1 + 0.414T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 - 1.39T + 47T^{2} \) |
| 53 | \( 1 - 2.83T + 53T^{2} \) |
| 59 | \( 1 + 0.759T + 59T^{2} \) |
| 61 | \( 1 + 4.75T + 61T^{2} \) |
| 67 | \( 1 - 0.735T + 67T^{2} \) |
| 71 | \( 1 - 6.89T + 71T^{2} \) |
| 73 | \( 1 + 6.38T + 73T^{2} \) |
| 79 | \( 1 - 3.13T + 79T^{2} \) |
| 83 | \( 1 + 8.49T + 83T^{2} \) |
| 89 | \( 1 + 0.954T + 89T^{2} \) |
| 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
−7.990884139911408732632119156716, −7.24654780543217737453760288551, −6.78252211347671605704178107174, −6.00516815403785662352514482742, −4.79817411314321392689792213157, −4.00707400917639401316012401868, −3.59928227063239551811407739686, −2.75672557269424832459539992141, −1.56729137109867320180361338505, −0.58837883180029228998453977245,
0.58837883180029228998453977245, 1.56729137109867320180361338505, 2.75672557269424832459539992141, 3.59928227063239551811407739686, 4.00707400917639401316012401868, 4.79817411314321392689792213157, 6.00516815403785662352514482742, 6.78252211347671605704178107174, 7.24654780543217737453760288551, 7.990884139911408732632119156716
