| L(s) = 1 | + 2.82i·3-s + 5-s − 5·7-s + 0.999·9-s + 5·11-s + 16.9i·13-s + 2.82i·15-s − 25·17-s + 19·19-s − 14.1i·21-s + 10·23-s − 24·25-s + 28.2i·27-s − 42.4i·29-s + 42.4i·31-s + ⋯ |
| L(s) = 1 | + 0.942i·3-s + 0.200·5-s − 0.714·7-s + 0.111·9-s + 0.454·11-s + 1.30i·13-s + 0.188i·15-s − 1.47·17-s + 19-s − 0.673i·21-s + 0.434·23-s − 0.959·25-s + 1.04i·27-s − 1.46i·29-s + 1.36i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9977659696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9977659696\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 2.82iT - 9T^{2} \) |
| 5 | \( 1 - T + 25T^{2} \) |
| 7 | \( 1 + 5T + 49T^{2} \) |
| 11 | \( 1 - 5T + 121T^{2} \) |
| 13 | \( 1 - 16.9iT - 169T^{2} \) |
| 17 | \( 1 + 25T + 289T^{2} \) |
| 23 | \( 1 - 10T + 529T^{2} \) |
| 29 | \( 1 + 42.4iT - 841T^{2} \) |
| 31 | \( 1 - 42.4iT - 961T^{2} \) |
| 37 | \( 1 - 25.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 42.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 25.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 84.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 95T + 3.72e3T^{2} \) |
| 67 | \( 1 - 110. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 25T + 5.32e3T^{2} \) |
| 79 | \( 1 + 42.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 130T + 6.88e3T^{2} \) |
| 89 | \( 1 + 127. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 16.9iT - 9.40e3T^{2} \) |
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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
−9.808961051413855264507718860053, −9.271442523655579667178009052352, −8.714721162518727063888217908963, −7.27818463600010053571397630899, −6.66607873069851729454915424837, −5.74087437482306725756663167257, −4.54049235092238358804179232279, −4.07753423488268528162614588511, −2.95769784078299026426650093159, −1.61890187183794563666153029922,
0.28877735576627156860567067410, 1.51464852629502267264975763343, 2.67701922849005728523121880777, 3.69752102300707023834843720612, 4.94968162690077241625855272886, 6.02618774718221267691993011872, 6.63279488846686857183789810806, 7.44798880492121281146743714858, 8.126668284886386582934411439794, 9.248721698815674170275888841848
