L(s) = 1 | + (5.44 + 5.44i)7-s + 6.44·11-s + (−14.4 + 14.4i)13-s + (−23.1 − 23.1i)17-s + 16.6i·19-s + (−6.65 + 6.65i)23-s + 0.0454i·29-s + 4.49·31-s + (−35.3 − 35.3i)37-s − 20.2·41-s + (−32.2 + 32.2i)43-s + (−50.5 − 50.5i)47-s + 10.3i·49-s + (−5.50 + 5.50i)53-s − 55.4i·59-s + ⋯ |
L(s) = 1 | + (0.778 + 0.778i)7-s + 0.586·11-s + (−1.11 + 1.11i)13-s + (−1.36 − 1.36i)17-s + 0.878i·19-s + (−0.289 + 0.289i)23-s + 0.00156i·29-s + 0.144·31-s + (−0.955 − 0.955i)37-s − 0.494·41-s + (−0.750 + 0.750i)43-s + (−1.07 − 1.07i)47-s + 0.212i·49-s + (−0.103 + 0.103i)53-s − 0.939i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2034426751\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2034426751\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-5.44 - 5.44i)T + 49iT^{2} \) |
| 11 | \( 1 - 6.44T + 121T^{2} \) |
| 13 | \( 1 + (14.4 - 14.4i)T - 169iT^{2} \) |
| 17 | \( 1 + (23.1 + 23.1i)T + 289iT^{2} \) |
| 19 | \( 1 - 16.6iT - 361T^{2} \) |
| 23 | \( 1 + (6.65 - 6.65i)T - 529iT^{2} \) |
| 29 | \( 1 - 0.0454iT - 841T^{2} \) |
| 31 | \( 1 - 4.49T + 961T^{2} \) |
| 37 | \( 1 + (35.3 + 35.3i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 + 20.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (32.2 - 32.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (50.5 + 50.5i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (5.50 - 5.50i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 55.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 47.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-85.2 - 85.2i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 48.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-21.9 + 21.9i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 126. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-94.9 + 94.9i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 71.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-37 - 37i)T + 9.40e3iT^{2} \) |
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
−9.366023161559868620101063134322, −8.844623228370468447664449880912, −8.002112849147330140089080843819, −7.04343881513526714319806500649, −6.48906939441477300097697302717, −5.24701295621894428009259561947, −4.77165234115165595819121431820, −3.74537719624819617674944962489, −2.37805854306248123029111828891, −1.75463237958958950307720057065,
0.04976566118830241009351476438, 1.39276645989792265415420596736, 2.48431590773195886236978996586, 3.71159077621917873505750563959, 4.57631691218532036584591948955, 5.21794485582512507503253009708, 6.46511046855768946764695522765, 7.01236431715650850267505961758, 8.028371878410979278516505193698, 8.464605862622837035395016275762