L(s) = 1 | + i·2-s − 4-s + (−1 − 2i)5-s + 5i·7-s − i·8-s + (2 − i)10-s + 11-s − 5·14-s + 16-s + 4i·17-s − 3·19-s + (1 + 2i)20-s + i·22-s − i·23-s + (−3 + 4i)25-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.447 − 0.894i)5-s + 1.88i·7-s − 0.353i·8-s + (0.632 − 0.316i)10-s + 0.301·11-s − 1.33·14-s + 0.250·16-s + 0.970i·17-s − 0.688·19-s + (0.223 + 0.447i)20-s + 0.213i·22-s − 0.208i·23-s + (−0.600 + 0.800i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5722747973\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5722747973\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| 31 | \( 1 + T \) |
good | 7 | \( 1 - 5iT - 7T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 + iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 + 14T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 + 15T + 79T^{2} \) |
| 83 | \( 1 + 10iT - 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.886219413233937066456271545131, −8.530706566840508756791543044039, −8.013839022563884743813875916976, −6.87405329908167991431624096935, −6.02428768007561862407224937574, −5.53799505196582219327969592858, −4.70715263235650874966324045221, −3.91643077326064205635575686963, −2.68473719931766097413205363077, −1.53702074080884874746533536959,
0.19353611109792239249739387624, 1.34960767249870833986809211380, 2.72925955331772984190899610736, 3.51240918904382389745026195888, 4.21683845573726497770047438271, 4.86628851339447821948924205768, 6.32715308451551047053990173459, 6.92782289578653014125952919763, 7.58832181191334339369297202948, 8.271669679684392021725819810208