L(s) = 1 | − 0.794i·3-s − 2.47i·7-s + 2.36·9-s − 2.29·11-s − 3.84i·13-s + 7.74i·17-s + 2.29·19-s − 1.96·21-s − i·23-s − 4.26i·27-s − 5.28·29-s − 6.40·31-s + 1.82i·33-s − 8.56i·37-s − 3.05·39-s + ⋯ |
L(s) = 1 | − 0.458i·3-s − 0.934i·7-s + 0.789·9-s − 0.693·11-s − 1.06i·13-s + 1.87i·17-s + 0.527·19-s − 0.428·21-s − 0.208i·23-s − 0.821i·27-s − 0.981·29-s − 1.14·31-s + 0.318i·33-s − 1.40i·37-s − 0.488·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.224049062\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.224049062\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 + 0.794iT - 3T^{2} \) |
| 7 | \( 1 + 2.47iT - 7T^{2} \) |
| 11 | \( 1 + 2.29T + 11T^{2} \) |
| 13 | \( 1 + 3.84iT - 13T^{2} \) |
| 17 | \( 1 - 7.74iT - 17T^{2} \) |
| 19 | \( 1 - 2.29T + 19T^{2} \) |
| 29 | \( 1 + 5.28T + 29T^{2} \) |
| 31 | \( 1 + 6.40T + 31T^{2} \) |
| 37 | \( 1 + 8.56iT - 37T^{2} \) |
| 41 | \( 1 - 4.27T + 41T^{2} \) |
| 43 | \( 1 + 1.88iT - 43T^{2} \) |
| 47 | \( 1 + 12.3iT - 47T^{2} \) |
| 53 | \( 1 + 7.57iT - 53T^{2} \) |
| 59 | \( 1 + 6.07T + 59T^{2} \) |
| 61 | \( 1 + 0.635T + 61T^{2} \) |
| 67 | \( 1 + 11.1iT - 67T^{2} \) |
| 71 | \( 1 - 8.58T + 71T^{2} \) |
| 73 | \( 1 - 16.5iT - 73T^{2} \) |
| 79 | \( 1 + 0.335T + 79T^{2} \) |
| 83 | \( 1 - 15.1iT - 83T^{2} \) |
| 89 | \( 1 + 5.55T + 89T^{2} \) |
| 97 | \( 1 + 6.42iT - 97T^{2} \) |
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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
−7.85771601162635669259925981186, −7.38194831997918020558356173014, −6.73349600961429717590409612948, −5.75734435216587769276954573466, −5.21696187022453887547214136448, −3.92894107326787477531994120286, −3.71653108994701176815663904300, −2.32245702870253218013955924472, −1.44781982575579223765552867465, −0.33643821475716445205698980173,
1.38683995820136882419082971015, 2.45599784625811697042424767458, 3.19572340252795069062776733344, 4.28689583780416074852469145202, 4.92798650439327093141302973704, 5.53645013995108366539649750176, 6.44406010646610696266542791880, 7.36714936971756997892216518668, 7.68037289148687569276358432320, 8.932101716087053894914602593073