L(s) = 1 | + 2.08i·5-s − 5.03·7-s − 0.828i·11-s − 2.94i·13-s − 4.82·17-s + 2.82i·19-s − 4.16·23-s + 0.656·25-s − 7.97i·29-s − 5.03·31-s − 10.4i·35-s + 7.11i·37-s + 8.82·41-s + 12.4i·43-s + 4.16·47-s + ⋯ |
L(s) = 1 | + 0.932i·5-s − 1.90·7-s − 0.249i·11-s − 0.817i·13-s − 1.17·17-s + 0.648i·19-s − 0.869·23-s + 0.131·25-s − 1.48i·29-s − 0.903·31-s − 1.77i·35-s + 1.16i·37-s + 1.37·41-s + 1.90i·43-s + 0.607·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9175234209\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9175234209\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2.08iT - 5T^{2} \) |
| 7 | \( 1 + 5.03T + 7T^{2} \) |
| 11 | \( 1 + 0.828iT - 11T^{2} \) |
| 13 | \( 1 + 2.94iT - 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 + 4.16T + 23T^{2} \) |
| 29 | \( 1 + 7.97iT - 29T^{2} \) |
| 31 | \( 1 + 5.03T + 31T^{2} \) |
| 37 | \( 1 - 7.11iT - 37T^{2} \) |
| 41 | \( 1 - 8.82T + 41T^{2} \) |
| 43 | \( 1 - 12.4iT - 43T^{2} \) |
| 47 | \( 1 - 4.16T + 47T^{2} \) |
| 53 | \( 1 + 12.1iT - 53T^{2} \) |
| 59 | \( 1 + 1.65iT - 59T^{2} \) |
| 61 | \( 1 + 7.11iT - 61T^{2} \) |
| 67 | \( 1 + 2.34iT - 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 5.03T + 79T^{2} \) |
| 83 | \( 1 + 3.17iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 0.343T + 97T^{2} \) |
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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
−8.213052772614479323489728757309, −7.54631485655975726947662223104, −6.63988014366753708683793329900, −6.28165114063932994727881201954, −5.72659621995582539921082742105, −4.40705363982815864757334252496, −3.54970026172621352009431139865, −2.99266904312020827319797213305, −2.22889121398405526136876112672, −0.44469414081934315029311288120,
0.57044281098716144365341057894, 1.98081251964077312322059457776, 2.88117916471366819731857548091, 3.93961769258695335163855434281, 4.40156212612306335432423895407, 5.51065545099226300603377625546, 6.09632481637188368336992444026, 7.05766763598886458734889342040, 7.22417458320238813252676039729, 8.714126009586454381326180467165