L(s) = 1 | − 2i·5-s − 4·7-s + 4i·11-s + 2i·13-s + 6·17-s − 4i·19-s + 25-s + 2i·29-s − 4·31-s + 8i·35-s − 2i·37-s + 2·41-s − 4i·43-s + 8·47-s + 9·49-s + ⋯ |
L(s) = 1 | − 0.894i·5-s − 1.51·7-s + 1.20i·11-s + 0.554i·13-s + 1.45·17-s − 0.917i·19-s + 0.200·25-s + 0.371i·29-s − 0.718·31-s + 1.35i·35-s − 0.328i·37-s + 0.312·41-s − 0.609i·43-s + 1.16·47-s + 1.28·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.352639401\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.352639401\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 10iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 16T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.196157733225529089312465358172, −8.206271792843165666601521568412, −7.16715810554002748224133427983, −6.77209989029074272953434809146, −5.67422571615892971359132728138, −4.97782031131378739858302163881, −4.02882779336633236721320512629, −3.17169361119004245545314740202, −1.99818597084615958384961295381, −0.62769668396498882839898116695,
0.888605342906179785623422702376, 2.70317677274366157697142455996, 3.26844259369493471845001666673, 3.87076273697347354838757659673, 5.53859079119232906435950130673, 5.95904907682833684919655114106, 6.69180965880791105959217480226, 7.54098490973968202557563080677, 8.236843162783280849131552804236, 9.261463209821477941125778635800