L(s) = 1 | + (−1 + 2i)5-s + 2i·7-s + 3·9-s + 4·11-s − 2i·13-s − 4i·17-s + 19-s + 6i·23-s + (−3 − 4i)25-s + 6·29-s + 4·31-s + (−4 − 2i)35-s + 10i·37-s − 10·41-s − 2i·43-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.894i)5-s + 0.755i·7-s + 9-s + 1.20·11-s − 0.554i·13-s − 0.970i·17-s + 0.229·19-s + 1.25i·23-s + (−0.600 − 0.800i)25-s + 1.11·29-s + 0.718·31-s + (−0.676 − 0.338i)35-s + 1.64i·37-s − 1.56·41-s − 0.304i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.791687987\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.791687987\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 18iT - 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−9.790974460603350459577491348606, −8.806023744950521977225630123010, −7.967209249590351443092100967014, −7.01972002348104482095709611608, −6.62479186634131935246910711123, −5.52323719735619434916334743381, −4.50484151717264425960352891776, −3.53009422507654082277128363168, −2.69093856588231215258699587545, −1.28701161864211654620757389450,
0.853342878682414649281281933757, 1.79689909352392986389703549877, 3.64562891955454834156087366130, 4.24941557729357117797711864981, 4.84547176461084993520862042114, 6.28470601196291565371436022421, 6.85927486385040849304503143387, 7.76788643969557361212284973950, 8.570691793726506884393445217455, 9.252687208463408578968196274943