L(s) = 1 | + 3.40i·3-s + 4.62i·7-s − 8.56·9-s + 0.223·13-s − 3.69·17-s − 4.35i·19-s − 15.7·21-s + 6.49i·23-s + 5·25-s − 18.9i·27-s + 6.57·29-s − 8.71·37-s + 0.761i·39-s + 6i·47-s − 14.4·49-s + ⋯ |
L(s) = 1 | + 1.96i·3-s + 1.74i·7-s − 2.85·9-s + 0.0620·13-s − 0.896·17-s − 0.999i·19-s − 3.43·21-s + 1.35i·23-s + 25-s − 3.64i·27-s + 1.22·29-s − 1.43·37-s + 0.121i·39-s + 0.875i·47-s − 2.06·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.072439141\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.072439141\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
good | 3 | \( 1 - 3.40iT - 3T^{2} \) |
| 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 4.62iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 0.223T + 13T^{2} \) |
| 17 | \( 1 + 3.69T + 17T^{2} \) |
| 23 | \( 1 - 6.49iT - 23T^{2} \) |
| 29 | \( 1 - 6.57T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 8.71T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 2.95iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 10.6iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 1.82T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−10.15053444550387956077660807171, −9.280111750085338042551482640704, −8.891186162271538954552144852330, −8.359052510903860639203916533728, −6.70413436449514002925873041512, −5.63329201267292235156841779203, −5.15907698055409035076454770711, −4.33983404975611072330396256558, −3.16629192483419579908779619337, −2.48814582241437693558831417997,
0.46324710390559103497528091274, 1.42381966829404287055727586117, 2.59854784271875324489770152520, 3.81352528800858615301340932305, 5.05089604853988102654477194056, 6.38124772592024266158399393760, 6.77962087857234738656363934211, 7.40664522805829684594495614224, 8.248610021536731023210375520929, 8.797101312446899810861241068814