| L(s) = 1 | − i·2-s + (−1.22 + 1.22i)3-s − 4-s + (1.22 + 1.22i)6-s + (1 − 2.44i)7-s + i·8-s − 2.99i·9-s + (1.22 − 1.22i)12-s + 2.44i·13-s + (−2.44 − i)14-s + 16-s − 4.89·17-s − 2.99·18-s + 2.44i·19-s + (1.77 + 4.22i)21-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + (−0.707 + 0.707i)3-s − 0.5·4-s + (0.499 + 0.499i)6-s + (0.377 − 0.925i)7-s + 0.353i·8-s − 0.999i·9-s + (0.353 − 0.353i)12-s + 0.679i·13-s + (−0.654 − 0.267i)14-s + 0.250·16-s − 1.18·17-s − 0.707·18-s + 0.561i·19-s + (0.387 + 0.921i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5480747325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5480747325\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1 + 2.44i)T \) |
| good | 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 4.89T + 17T^{2} \) |
| 19 | \( 1 - 2.44iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 4.89T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4.89T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 12.2iT - 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 2.44T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.852819731215893721979126525747, −8.962877995483475620472544634756, −8.089589908124470474488669096912, −6.84498733593176397322013507869, −6.11871470584013594334326450483, −4.71388457137282644029072364478, −4.41928024822430936023081206773, −3.38507631863004167325127807573, −1.83621883211207400100821098209, −0.27259851216497968689138301265,
1.55682359180407164960799511877, 2.91252501036235362599200584448, 4.58833020679123276816062157146, 5.33583705371475044487665834192, 6.01828880890777802479482610364, 6.89799243615811870020358748348, 7.63779917557935518044545338502, 8.507228337297251442255793586739, 9.163101642626970280542953369970, 10.33011665064657163968570235294
