L(s) = 1 | + (3.48 − 6.04i)5-s − 3.44·7-s + 10.1·11-s + (−0.151 + 0.0874i)13-s + (1.56 − 2.71i)17-s + (11.3 − 15.2i)19-s + (3.48 + 6.04i)23-s + (−11.8 − 20.5i)25-s + (−4.70 + 2.71i)29-s − 36.8i·31-s + (−12.0 + 20.8i)35-s − 27.1i·37-s + (−51.2 − 29.6i)41-s + (10.9 − 19.0i)43-s + (6.27 + 10.8i)47-s + ⋯ |
L(s) = 1 | + (0.697 − 1.20i)5-s − 0.492·7-s + 0.919·11-s + (−0.0116 + 0.00672i)13-s + (0.0922 − 0.159i)17-s + (0.597 − 0.802i)19-s + (0.151 + 0.262i)23-s + (−0.473 − 0.820i)25-s + (−0.162 + 0.0936i)29-s − 1.18i·31-s + (−0.343 + 0.595i)35-s − 0.734i·37-s + (−1.25 − 0.722i)41-s + (0.255 − 0.441i)43-s + (0.133 + 0.231i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0186 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0186 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.876253348\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.876253348\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-11.3 + 15.2i)T \) |
good | 5 | \( 1 + (-3.48 + 6.04i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + 3.44T + 49T^{2} \) |
| 11 | \( 1 - 10.1T + 121T^{2} \) |
| 13 | \( 1 + (0.151 - 0.0874i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-1.56 + 2.71i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + (-3.48 - 6.04i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (4.70 - 2.71i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + 36.8iT - 961T^{2} \) |
| 37 | \( 1 + 27.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (51.2 + 29.6i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-10.9 + 19.0i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-6.27 - 10.8i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (36.1 - 20.8i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-21.9 - 12.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (26.1 + 45.2i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-41.3 + 23.8i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-14.1 - 8.14i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-14.9 + 25.8i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (53.1 + 30.7i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 148.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-54.9 + 31.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-27 - 15.5i)T + (4.70e3 + 8.14e3i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.654545451693880276263556129629, −9.360466500853280667971840880758, −8.577533415172292310351098969181, −7.40625146102387951796917031690, −6.39271177830041892676827018234, −5.50055238151727439826115812846, −4.63757081478656568777585039103, −3.46190443158437838916187159338, −1.94074133881091418927663379065, −0.69847906787698184686371475923,
1.53717491798908231660460952428, 2.89741487280811206721813905389, 3.69620568110846372210615494531, 5.17479331099076940012974939984, 6.39882119277928507687570766503, 6.62431723279747100663719450413, 7.78395284027341119509683445826, 8.921341043391507917980520536933, 9.863114120911182500633052158483, 10.28806634521620971294769502062