L(s) = 1 | + (0.671 + 0.740i)2-s + (−0.0980 + 0.995i)4-s + (−0.146 − 0.989i)7-s + (−0.803 + 0.595i)8-s + (0.514 − 0.857i)9-s + (0.0153 − 0.0466i)11-s + (0.634 − 0.773i)14-s + (−0.980 − 0.195i)16-s + (0.980 − 0.195i)18-s + (0.0448 − 0.0198i)22-s + (0.326 − 0.360i)23-s + (0.336 + 0.941i)25-s + (0.998 − 0.0490i)28-s + (1.96 + 0.144i)29-s + (−0.514 − 0.857i)32-s + ⋯ |
L(s) = 1 | + (0.671 + 0.740i)2-s + (−0.0980 + 0.995i)4-s + (−0.146 − 0.989i)7-s + (−0.803 + 0.595i)8-s + (0.514 − 0.857i)9-s + (0.0153 − 0.0466i)11-s + (0.634 − 0.773i)14-s + (−0.980 − 0.195i)16-s + (0.980 − 0.195i)18-s + (0.0448 − 0.0198i)22-s + (0.326 − 0.360i)23-s + (0.336 + 0.941i)25-s + (0.998 − 0.0490i)28-s + (1.96 + 0.144i)29-s + (−0.514 − 0.857i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.839487646\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.839487646\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.671 - 0.740i)T \) |
| 7 | \( 1 + (0.146 + 0.989i)T \) |
good | 3 | \( 1 + (-0.514 + 0.857i)T^{2} \) |
| 5 | \( 1 + (-0.336 - 0.941i)T^{2} \) |
| 11 | \( 1 + (-0.0153 + 0.0466i)T + (-0.803 - 0.595i)T^{2} \) |
| 13 | \( 1 + (0.427 + 0.903i)T^{2} \) |
| 17 | \( 1 + (0.980 - 0.195i)T^{2} \) |
| 19 | \( 1 + (0.998 - 0.0490i)T^{2} \) |
| 23 | \( 1 + (-0.326 + 0.360i)T + (-0.0980 - 0.995i)T^{2} \) |
| 29 | \( 1 + (-1.96 - 0.144i)T + (0.989 + 0.146i)T^{2} \) |
| 31 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 37 | \( 1 + (-0.585 + 0.225i)T + (0.740 - 0.671i)T^{2} \) |
| 41 | \( 1 + (0.773 + 0.634i)T^{2} \) |
| 43 | \( 1 + (-1.45 + 0.403i)T + (0.857 - 0.514i)T^{2} \) |
| 47 | \( 1 + (0.831 + 0.555i)T^{2} \) |
| 53 | \( 1 + (-0.244 + 0.0180i)T + (0.989 - 0.146i)T^{2} \) |
| 59 | \( 1 + (-0.427 + 0.903i)T^{2} \) |
| 61 | \( 1 + (-0.242 - 0.970i)T^{2} \) |
| 67 | \( 1 + (-0.0652 - 0.529i)T + (-0.970 + 0.242i)T^{2} \) |
| 71 | \( 1 + (-0.229 + 0.914i)T + (-0.881 - 0.471i)T^{2} \) |
| 73 | \( 1 + (0.956 + 0.290i)T^{2} \) |
| 79 | \( 1 + (1.59 + 0.852i)T + (0.555 + 0.831i)T^{2} \) |
| 83 | \( 1 + (-0.740 - 0.671i)T^{2} \) |
| 89 | \( 1 + (0.0980 - 0.995i)T^{2} \) |
| 97 | \( 1 + (0.382 - 0.923i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.725753152646468640716660858843, −7.80791678783340249153498303072, −7.16382360664405487742630395524, −6.63394711326636147547259061521, −5.96496036582345843880693522730, −4.91040177336610898492664912486, −4.26252467991949690013935338400, −3.56742879540292981985154188598, −2.72064053943493505687597323328, −1.03471089610191301461752851008,
1.26873811405448189861541511631, 2.43783243244713883449192797527, 2.85970475046393605644915370275, 4.15277085898876820160131450466, 4.75577545219503652458472289636, 5.50222834003325611571265214555, 6.24686904427773216025084707036, 6.98373379364895191613883799211, 8.082931919532862680503121729027, 8.732117642487740214916467730628