L(s) = 1 | + 2.90i·3-s − 3.66·5-s − 1.93i·7-s + 0.583·9-s + 0.752i·11-s + 13.0·13-s − 10.6i·15-s − 23.9·17-s − 4.35i·19-s + 5.61·21-s + 6.26i·23-s − 11.5·25-s + 27.8i·27-s − 33.1·29-s − 17.5i·31-s + ⋯ |
L(s) = 1 | + 0.967i·3-s − 0.732·5-s − 0.276i·7-s + 0.0647·9-s + 0.0684i·11-s + 1.00·13-s − 0.708i·15-s − 1.40·17-s − 0.229i·19-s + 0.267·21-s + 0.272i·23-s − 0.463·25-s + 1.02i·27-s − 1.14·29-s − 0.565i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5685243724\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5685243724\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
good | 3 | \( 1 - 2.90iT - 9T^{2} \) |
| 5 | \( 1 + 3.66T + 25T^{2} \) |
| 7 | \( 1 + 1.93iT - 49T^{2} \) |
| 11 | \( 1 - 0.752iT - 121T^{2} \) |
| 13 | \( 1 - 13.0T + 169T^{2} \) |
| 17 | \( 1 + 23.9T + 289T^{2} \) |
| 23 | \( 1 - 6.26iT - 529T^{2} \) |
| 29 | \( 1 + 33.1T + 841T^{2} \) |
| 31 | \( 1 + 17.5iT - 961T^{2} \) |
| 37 | \( 1 + 41.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 5.51T + 1.68e3T^{2} \) |
| 43 | \( 1 + 84.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 18.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 41.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 69.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 87.6T + 3.72e3T^{2} \) |
| 67 | \( 1 + 105. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 74.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 48.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 95.9iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 65.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 3.38T + 7.92e3T^{2} \) |
| 97 | \( 1 + 23.5T + 9.40e3T^{2} \) |
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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
−9.147233197551487138339191657411, −8.835046170202786034253820415603, −7.66463989638124156243538232924, −6.97434436721230797555528667651, −5.87761657368754167880508654504, −4.83011551764173562050103019631, −3.98203192881268993167842681864, −3.54979363700854474627531899033, −1.91148213145747218196168648578, −0.17527396234169597570187999478,
1.25018188425621450320873991736, 2.31140424854855576652155120204, 3.63719213518055234555273105676, 4.46533652293891517457625700917, 5.74990135760513622207527036149, 6.58555650448898716525352212728, 7.23549799879356778252068244736, 8.119098967224661068176317564721, 8.666244430177619196740365541403, 9.630662126253883794523567499059