| L(s) = 1 | − 2.56i·5-s − i·7-s − 1.15·11-s + 0.578·13-s + 5.39i·17-s + 6.20i·19-s + 7.62·23-s − 1.57·25-s + 1.41i·29-s + 5.04i·31-s − 2.56·35-s − 9.83·37-s − 6.21i·41-s + 11.2i·43-s − 11.0·47-s + ⋯ |
| L(s) = 1 | − 1.14i·5-s − 0.377i·7-s − 0.346·11-s + 0.160·13-s + 1.30i·17-s + 1.42i·19-s + 1.58·23-s − 0.315·25-s + 0.262i·29-s + 0.906i·31-s − 0.433·35-s − 1.61·37-s − 0.970i·41-s + 1.71i·43-s − 1.61·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.556913165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.556913165\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 + 2.56iT - 5T^{2} \) |
| 11 | \( 1 + 1.15T + 11T^{2} \) |
| 13 | \( 1 - 0.578T + 13T^{2} \) |
| 17 | \( 1 - 5.39iT - 17T^{2} \) |
| 19 | \( 1 - 6.20iT - 19T^{2} \) |
| 23 | \( 1 - 7.62T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 5.04iT - 31T^{2} \) |
| 37 | \( 1 + 9.83T + 37T^{2} \) |
| 41 | \( 1 + 6.21iT - 41T^{2} \) |
| 43 | \( 1 - 11.2iT - 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 4.53iT - 53T^{2} \) |
| 59 | \( 1 - 4.83T + 59T^{2} \) |
| 61 | \( 1 + 0.951T + 61T^{2} \) |
| 67 | \( 1 + 2.78iT - 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 - 12.8iT - 79T^{2} \) |
| 83 | \( 1 - 8.77T + 83T^{2} \) |
| 89 | \( 1 - 5.68iT - 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.389512156897243121261599821040, −8.069046817809705471820544099422, −7.04021333504557652311742856131, −6.30459908708859962003581518211, −5.33927123085062869699311475463, −4.90656352349358862332723731601, −3.91747364297512394945235982032, −3.23605205487389748954127461677, −1.76824760168611945423313129837, −1.07532066234199239964883932544,
0.50113396288982743828237085438, 2.14326820126507998570725508745, 2.89077657299465332597075922509, 3.45790715269797027894047300192, 4.83345936603194771832575835671, 5.23296238309964372868419066909, 6.37496189731986800320986297257, 6.97339208594911334774738698233, 7.37365576788897054138911530296, 8.431452202451274676153990428566
