L(s) = 1 | + i·2-s + i·3-s − 4-s − 0.357i·5-s − 6-s − i·8-s − 9-s + 0.357·10-s + (−3.31 + 0.129i)11-s − i·12-s + 4.33·13-s + 0.357·15-s + 16-s + 0.347·17-s − i·18-s − 7.81·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.159i·5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + 0.112·10-s + (−0.999 + 0.0389i)11-s − 0.288i·12-s + 1.20·13-s + 0.0922·15-s + 0.250·16-s + 0.0842·17-s − 0.235i·18-s − 1.79·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.420105821\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.420105821\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (3.31 - 0.129i)T \) |
good | 5 | \( 1 + 0.357iT - 5T^{2} \) |
| 13 | \( 1 - 4.33T + 13T^{2} \) |
| 17 | \( 1 - 0.347T + 17T^{2} \) |
| 19 | \( 1 + 7.81T + 19T^{2} \) |
| 23 | \( 1 - 6.36T + 23T^{2} \) |
| 29 | \( 1 + 7.67iT - 29T^{2} \) |
| 31 | \( 1 + 6.20iT - 31T^{2} \) |
| 37 | \( 1 + 8.95T + 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 - 2.91iT - 43T^{2} \) |
| 47 | \( 1 + 3.84iT - 47T^{2} \) |
| 53 | \( 1 + 2.73T + 53T^{2} \) |
| 59 | \( 1 + 0.911iT - 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 6.18T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 6.65iT - 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 + 14.5iT - 89T^{2} \) |
| 97 | \( 1 + 18.1iT - 97T^{2} \) |
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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
−8.525075481833505927595164386910, −8.203346628851156705991816816026, −7.18918348330808682570808292445, −6.35505191686144383510677785121, −5.73260944999893797505651931949, −4.88340163478589289773854788825, −4.23381799846769837250319192805, −3.33221108556385214202294560127, −2.21284641511540281032999500538, −0.55201754387237287577678545830,
0.932623548264701062941193283024, 1.95819305447583390514405833697, 2.92733412803787526943369013591, 3.61020019494911516573722509267, 4.79642263853537133682589387974, 5.42145990953975656600194152100, 6.49160333768962290806515394865, 7.00305715782183412778117191079, 8.064227875662871787813604276052, 8.702259022823307434835539979507