| L(s) = 1 | + i·3-s − 5-s + (1.73 + 2i)7-s + 2·9-s − 5.19·11-s − 13-s − i·15-s + 1.73i·17-s + 2i·19-s + (−2 + 1.73i)21-s + 25-s + 5i·27-s − 1.73i·29-s − 3.46·31-s − 5.19i·33-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447·5-s + (0.654 + 0.755i)7-s + 0.666·9-s − 1.56·11-s − 0.277·13-s − 0.258i·15-s + 0.420i·17-s + 0.458i·19-s + (−0.436 + 0.377i)21-s + 0.200·25-s + 0.962i·27-s − 0.321i·29-s − 0.622·31-s − 0.904i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7630540298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7630540298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-1.73 - 2i)T \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 11 | \( 1 + 5.19T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 1.73iT - 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 1.73iT - 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 6.92iT - 37T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 + 12.1T + 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - iT - 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 8.66iT - 97T^{2} \) |
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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
−9.534307423831370359945232687332, −8.505808707624431138472033960724, −7.949905056115618222239845943481, −7.32712882661550949329331085280, −6.15798440911571505444892562878, −5.20783487621310378211689802801, −4.76367603029502153312594453892, −3.77378442789939393309484733500, −2.74811780539562572962338969934, −1.68820224036392088816101582833,
0.25539979652445907071838766441, 1.54680270585213125388423708404, 2.62907537950430314082703768913, 3.74754499604819816080202944143, 4.79162632323330387760359274967, 5.22843328812613373151057911881, 6.62353254473717512154210860343, 7.22564702038750294341780081951, 7.83075261238603487709329810340, 8.310154399297307237021366203934
