L(s) = 1 | + (1.58 − 0.707i)3-s + 1.41·5-s + (2.23 − 1.41i)7-s + (2.00 − 2.23i)9-s + 3.16i·13-s + (2.23 − 1.00i)15-s − 2.82·17-s − 4.24i·19-s + (2.53 − 3.81i)21-s + 6i·23-s − 2.99·25-s + (1.58 − 4.94i)27-s + 4.47i·29-s − 5.65i·31-s + (3.16 − 2.00i)35-s + ⋯ |
L(s) = 1 | + (0.912 − 0.408i)3-s + 0.632·5-s + (0.845 − 0.534i)7-s + (0.666 − 0.745i)9-s + 0.877i·13-s + (0.577 − 0.258i)15-s − 0.685·17-s − 0.973i·19-s + (0.553 − 0.832i)21-s + 1.25i·23-s − 0.599·25-s + (0.304 − 0.952i)27-s + 0.830i·29-s − 1.01i·31-s + (0.534 − 0.338i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.31191 - 0.697867i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.31191 - 0.697867i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.58 + 0.707i)T \) |
| 7 | \( 1 + (-2.23 + 1.41i)T \) |
good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 3.16iT - 13T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 + 4.24iT - 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 4.47iT - 29T^{2} \) |
| 31 | \( 1 + 5.65iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 8.48T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 - 6.32T + 47T^{2} \) |
| 53 | \( 1 - 13.4iT - 53T^{2} \) |
| 59 | \( 1 + 3.16T + 59T^{2} \) |
| 61 | \( 1 - 3.16iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 8iT - 71T^{2} \) |
| 73 | \( 1 + 12.6iT - 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 - 3.16T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 18.9iT - 97T^{2} \) |
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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
−10.34785232543831976467587206815, −9.264140364354715661214163260257, −8.904850080288182023142212343316, −7.67857467875216776839741900938, −7.14358143904882986017787000807, −6.07715985868060560534161140201, −4.76588791564459805278675929609, −3.82362419969975428280808564303, −2.39382602341030359544141286169, −1.45977132553862462958433519583,
1.80306740227037151909350174751, 2.71653512446063766474166715921, 4.04049920909941846092705967937, 5.06429803855521743328566526755, 5.95724100138025487036469149635, 7.25397862630324803634168619093, 8.353845437921543701180375920052, 8.610555988265087813553779581630, 9.811457260887275917191515325298, 10.34198868652015159786968967879