| L(s) = 1 | + (1.22 + 1.22i)3-s + 2.44·5-s + i·7-s + 2.99i·9-s + 2.44i·13-s + (2.99 + 2.99i)15-s + 6i·17-s − 7.34·19-s + (−1.22 + 1.22i)21-s + 0.999·25-s + (−3.67 + 3.67i)27-s + 4.89·29-s − 8i·31-s + 2.44i·35-s + 4.89i·37-s + ⋯ |
| L(s) = 1 | + (0.707 + 0.707i)3-s + 1.09·5-s + 0.377i·7-s + 0.999i·9-s + 0.679i·13-s + (0.774 + 0.774i)15-s + 1.45i·17-s − 1.68·19-s + (−0.267 + 0.267i)21-s + 0.199·25-s + (−0.707 + 0.707i)27-s + 0.909·29-s − 1.43i·31-s + 0.414i·35-s + 0.805i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.417065390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.417065390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 - 2.44T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 4.89T + 29T^{2} \) |
| 31 | \( 1 + 8iT - 31T^{2} \) |
| 37 | \( 1 - 4.89iT - 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 - 4.89T + 43T^{2} \) |
| 47 | \( 1 - 12T + 47T^{2} \) |
| 53 | \( 1 - 9.79T + 53T^{2} \) |
| 59 | \( 1 - 2.44iT - 59T^{2} \) |
| 61 | \( 1 + 2.44iT - 61T^{2} \) |
| 67 | \( 1 + 9.79T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 8iT - 79T^{2} \) |
| 83 | \( 1 - 7.34iT - 83T^{2} \) |
| 89 | \( 1 + 18iT - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.877953973342625506241594554782, −8.797638896315111526209825952057, −8.684696303006516167227922963639, −7.50340461179847692891810876849, −6.25179397249292919492592124636, −5.79666651294002561796875171556, −4.54599955040675263023024731308, −3.88942931954054971697813552874, −2.46220368988041504503211971374, −1.90677320669866998330697495805,
0.905407052361729299583494746936, 2.19896121790962692168036762269, 2.89860909753782316651298128304, 4.18308477610880288272149520959, 5.34552400088163909847411344338, 6.26730928262159633670714046618, 6.95053400429764121749539875574, 7.75944359984751048321146973417, 8.716043919284907837689313091008, 9.243260921690765642757303188088
