| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s − 2·13-s + 15-s − 4·17-s − 19-s + 2·21-s + 23-s + 25-s − 27-s + 6·29-s − 2·31-s + 2·35-s + 2·37-s + 2·39-s + 12·41-s − 45-s + 8·47-s − 3·49-s + 4·51-s + 2·53-s + 57-s + 6·59-s − 2·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s − 0.970·17-s − 0.229·19-s + 0.436·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s + 0.338·35-s + 0.328·37-s + 0.320·39-s + 1.87·41-s − 0.149·45-s + 1.16·47-s − 3/7·49-s + 0.560·51-s + 0.274·53-s + 0.132·57-s + 0.781·59-s − 0.256·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.90687901686623, −13.23266074169917, −13.00850892326549, −12.44739974963215, −11.99106408405814, −11.62045895264088, −10.90141284447019, −10.65093029245176, −10.12494370904749, −9.396494756393101, −9.197951201274959, −8.487023744332943, −7.952344361479438, −7.219221784824966, −7.027017259710894, −6.306204134569117, −5.975741093856041, −5.281212048024172, −4.635693145188908, −4.216272953311185, −3.694891318934848, −2.708473251350241, −2.566693433712420, −1.485040049046084, −0.6772092708831887, 0,
0.6772092708831887, 1.485040049046084, 2.566693433712420, 2.708473251350241, 3.694891318934848, 4.216272953311185, 4.635693145188908, 5.281212048024172, 5.975741093856041, 6.306204134569117, 7.027017259710894, 7.219221784824966, 7.952344361479438, 8.487023744332943, 9.197951201274959, 9.396494756393101, 10.12494370904749, 10.65093029245176, 10.90141284447019, 11.62045895264088, 11.99106408405814, 12.44739974963215, 13.00850892326549, 13.23266074169917, 13.90687901686623
