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
−8.932101716087053894914602593073, −7.68037289148687569276358432320, −7.36714936971756997892216518668, −6.44406010646610696266542791880, −5.53645013995108366539649750176, −4.92798650439327093141302973704, −4.28689583780416074852469145202, −3.19572340252795069062776733344, −2.45599784625811697042424767458, −1.38683995820136882419082971015,
0.33643821475716445205698980173, 1.44781982575579223765552867465, 2.32245702870253218013955924472, 3.71653108994701176815663904300, 3.92894107326787477531994120286, 5.21696187022453887547214136448, 5.75734435216587769276954573466, 6.73349600961429717590409612948, 7.38194831997918020558356173014, 7.85771601162635669259925981186