| L(s) = 1 | − i·3-s − 2.88·5-s − 9-s + 5.82·11-s + 1.04·13-s + 2.88i·15-s + 6.82i·17-s + 0.681i·19-s + 2.14i·23-s + 3.31·25-s + i·27-s − 6.61i·29-s − 3.83·31-s − 5.82i·33-s − 2.38i·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.28·5-s − 0.333·9-s + 1.75·11-s + 0.290·13-s + 0.744i·15-s + 1.65i·17-s + 0.156i·19-s + 0.447i·23-s + 0.662·25-s + 0.192i·27-s − 1.22i·29-s − 0.688·31-s − 1.01i·33-s − 0.391i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0215 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0215 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8246919668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8246919668\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2.88T + 5T^{2} \) |
| 11 | \( 1 - 5.82T + 11T^{2} \) |
| 13 | \( 1 - 1.04T + 13T^{2} \) |
| 17 | \( 1 - 6.82iT - 17T^{2} \) |
| 19 | \( 1 - 0.681iT - 19T^{2} \) |
| 23 | \( 1 - 2.14iT - 23T^{2} \) |
| 29 | \( 1 + 6.61iT - 29T^{2} \) |
| 31 | \( 1 + 3.83T + 31T^{2} \) |
| 37 | \( 1 + 2.38iT - 37T^{2} \) |
| 41 | \( 1 - 1.19iT - 41T^{2} \) |
| 43 | \( 1 + 1.34T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 8.07iT - 53T^{2} \) |
| 59 | \( 1 - 7.87iT - 59T^{2} \) |
| 61 | \( 1 + 3.26T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 - 1.08iT - 71T^{2} \) |
| 73 | \( 1 + 5.64iT - 73T^{2} \) |
| 79 | \( 1 - 12.6iT - 79T^{2} \) |
| 83 | \( 1 - 0.482iT - 83T^{2} \) |
| 89 | \( 1 - 12.3iT - 89T^{2} \) |
| 97 | \( 1 - 3.63iT - 97T^{2} \) |
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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
−8.314080707361970119267028666874, −7.84794717424954719478499716088, −7.06032908895972323257641500918, −6.37431570248956327996365224068, −5.82582903055321035653408305326, −4.53518906340647821758457827795, −3.81741328953332961932550205719, −3.45938236907843355992081809980, −1.95114939028123364926994130670, −1.12361275119277606623811426221,
0.25964689008620198963277978235, 1.50126315265522180812776033257, 3.07315546801811073905437638394, 3.52694131298527918303308596033, 4.42321121715167370918583474620, 4.83294554677113151455340425621, 5.96311649117339208059480831854, 6.84848836103871110357739756978, 7.29919563904417259322236733745, 8.191816798850373697964758309629
