| L(s) = 1 | + i·2-s + i·3-s − 4-s − 6-s − 4i·7-s − i·8-s − 9-s + 2·11-s − i·12-s + 2i·13-s + 4·14-s + 16-s − i·18-s + 4·21-s + 2i·22-s − 6i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.51i·7-s − 0.353i·8-s − 0.333·9-s + 0.603·11-s − 0.288i·12-s + 0.554i·13-s + 1.06·14-s + 0.250·16-s − 0.235i·18-s + 0.872·21-s + 0.426i·22-s − 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9875924758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9875924758\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 16T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 8iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.272123201269666168822502921265, −7.22411680939721347611360175952, −6.87140230057815804186830204310, −6.13835796015377717665060690712, −5.14483159232584940169748766426, −4.31280744340401531876179352982, −4.02688514323385123971204441249, −3.02044061579334669816179694053, −1.50754747949803196528369502146, −0.28328791655366484892344175710,
1.26261788932869135444851957440, 2.08217145926231819182240236829, 2.92025008839321598781830201515, 3.62070270495261969137166127739, 4.81450535589317573670493627662, 5.55120044252571257043694027319, 6.08624236532244518079889409987, 6.99744635932229683650727294650, 7.910612309692671675574872999370, 8.525066850308287607996322717318
