L(s) = 1 | + (−1.07 + 0.921i)2-s + (0.303 − 1.97i)4-s + i·5-s + (−1.82 + 1.91i)7-s + (1.49 + 2.40i)8-s + (−0.921 − 1.07i)10-s + 6.24i·11-s − 2.40i·13-s + (0.195 − 3.73i)14-s + (−3.81 − 1.19i)16-s + 1.30i·17-s − 3.94·19-s + (1.97 + 0.303i)20-s + (−5.74 − 6.69i)22-s − 3.55i·23-s + ⋯ |
L(s) = 1 | + (−0.758 + 0.651i)2-s + (0.151 − 0.988i)4-s + 0.447i·5-s + (−0.690 + 0.723i)7-s + (0.528 + 0.848i)8-s + (−0.291 − 0.339i)10-s + 1.88i·11-s − 0.666i·13-s + (0.0522 − 0.998i)14-s + (−0.954 − 0.299i)16-s + 0.317i·17-s − 0.905·19-s + (0.442 + 0.0677i)20-s + (−1.22 − 1.42i)22-s − 0.741i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 + 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3429099979\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3429099979\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.07 - 0.921i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (1.82 - 1.91i)T \) |
good | 11 | \( 1 - 6.24iT - 11T^{2} \) |
| 13 | \( 1 + 2.40iT - 13T^{2} \) |
| 17 | \( 1 - 1.30iT - 17T^{2} \) |
| 19 | \( 1 + 3.94T + 19T^{2} \) |
| 23 | \( 1 + 3.55iT - 23T^{2} \) |
| 29 | \( 1 + 1.44T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 + 5.06T + 37T^{2} \) |
| 41 | \( 1 + 1.26iT - 41T^{2} \) |
| 43 | \( 1 - 2.19iT - 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 + 0.415T + 59T^{2} \) |
| 61 | \( 1 + 12.6iT - 61T^{2} \) |
| 67 | \( 1 - 3.10iT - 67T^{2} \) |
| 71 | \( 1 - 10.4iT - 71T^{2} \) |
| 73 | \( 1 + 1.64iT - 73T^{2} \) |
| 79 | \( 1 - 9.18iT - 79T^{2} \) |
| 83 | \( 1 + 7.39T + 83T^{2} \) |
| 89 | \( 1 + 11.4iT - 89T^{2} \) |
| 97 | \( 1 - 12.4iT - 97T^{2} \) |
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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
−9.993874455856066630075189025208, −9.485012249976236327549064223210, −8.442701291136009190783497252108, −7.83805182545097418458558522917, −6.63634171756331439113360191830, −6.53025739266478436823345265905, −5.29058997376771331841159685614, −4.41537713394945001098033970117, −2.80811339429930452825710185865, −1.85422872506046728299602927891,
0.18900517124329574358531436916, 1.37433869576112744664532976990, 2.92063470765350886182836755447, 3.66356963159417691733972611854, 4.64238498711069521137982658047, 6.10522522861294387626976664047, 6.76335071956734248011774168349, 7.87109854752387849347291182680, 8.522375790502496869221758430019, 9.219180614046434768287312473964