L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + 3.46·7-s + 2.82·8-s + 2.82i·11-s + 3.46·13-s + (−2.44 − 4.24i)14-s + (−2.00 − 3.46i)16-s − 1.41·17-s + 4·19-s + (3.46 − 2.00i)22-s + 4.89i·23-s + (−2.44 − 4.24i)26-s + (−3.46 + 5.99i)28-s − 2.44·29-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 1.30·7-s + 0.999·8-s + 0.852i·11-s + 0.960·13-s + (−0.654 − 1.13i)14-s + (−0.500 − 0.866i)16-s − 0.342·17-s + 0.917·19-s + (0.738 − 0.426i)22-s + 1.02i·23-s + (−0.480 − 0.832i)26-s + (−0.654 + 1.13i)28-s − 0.454·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.541109225\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.541109225\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 4.89iT - 23T^{2} \) |
| 29 | \( 1 + 2.44T + 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 4.89iT - 47T^{2} \) |
| 53 | \( 1 - 7.34iT - 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 13.8iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 14.6T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.284975654208881976895777466884, −8.616280552188135757209829109581, −7.69718253128671125798305661466, −7.38616793284756931115814949093, −5.94401773592636924474191711123, −4.88585896679264047759227292340, −4.23421106219327509888311390362, −3.19847731606010655949609777588, −1.96375948606324944309882074572, −1.22427919871319171556037843505,
0.820413792831509259468289644955, 1.91502674844665322562019872667, 3.55134353859166349524417604376, 4.63607026990518304485101258983, 5.35913351615737614647606944364, 6.11930714493494465697630371291, 7.00599493579714642493072078442, 7.83675991308938237292913454584, 8.553611860552810907866831326171, 8.854948075133559271354024203297