| L(s) = 1 | + 1.73i·3-s + i·5-s + 7-s − i·11-s + i·13-s − 1.73·15-s − 5.73·17-s − 2i·19-s + 1.73i·21-s − 3.46·23-s − 25-s + 5.19i·27-s + 9.19i·29-s + 4·31-s + 1.73·33-s + ⋯ |
| L(s) = 1 | + 0.999i·3-s + 0.447i·5-s + 0.377·7-s − 0.301i·11-s + 0.277i·13-s − 0.447·15-s − 1.39·17-s − 0.458i·19-s + 0.377i·21-s − 0.722·23-s − 0.200·25-s + 1.00i·27-s + 1.70i·29-s + 0.718·31-s + 0.301·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.207343952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.207343952\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - T \) |
| good | 3 | \( 1 - 1.73iT - 3T^{2} \) |
| 11 | \( 1 + iT - 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 + 5.73T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 9.19iT - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 7.46iT - 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 3.92T + 47T^{2} \) |
| 53 | \( 1 - 5.46iT - 53T^{2} \) |
| 59 | \( 1 + 2.53iT - 59T^{2} \) |
| 61 | \( 1 - 5.46iT - 61T^{2} \) |
| 67 | \( 1 + 1.46iT - 67T^{2} \) |
| 71 | \( 1 - 0.535T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 5.73T + 79T^{2} \) |
| 83 | \( 1 + 15.4iT - 83T^{2} \) |
| 89 | \( 1 - 9.46T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−9.350696551263501935359867666509, −8.787055489206221612954602807383, −7.977844650186741585147496109170, −6.89512796608448601502474913199, −6.41123199012505121516792024927, −5.13194718548715802490333560504, −4.63117398905761460211304919075, −3.75386345304978171782149845304, −2.86135888640375600816115329601, −1.61926641243273537040398930121,
0.39939217807741391089895148624, 1.73390090913425593696017124909, 2.34687615563925598297920375453, 3.89614781438349224829597868743, 4.62004931163814357376851604413, 5.63020294566559132902020508002, 6.48991085905297438331229138434, 7.08940167251345399022332748979, 8.116914156021555648146572828393, 8.273533810786935791409564051690
