| L(s) = 1 | − 7.50i·2-s + 17.3i·3-s − 24.3·4-s + 130.·6-s + 149. i·7-s − 57.2i·8-s − 58.1·9-s − 492.·11-s − 423. i·12-s − 169i·13-s + 1.12e3·14-s − 1.20e3·16-s − 1.82e3i·17-s + 436. i·18-s − 424.·19-s + ⋯ |
| L(s) = 1 | − 1.32i·2-s + 1.11i·3-s − 0.761·4-s + 1.47·6-s + 1.15i·7-s − 0.316i·8-s − 0.239·9-s − 1.22·11-s − 0.848i·12-s − 0.277i·13-s + 1.53·14-s − 1.18·16-s − 1.53i·17-s + 0.317i·18-s − 0.269·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.864319314\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.864319314\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 + 169iT \) |
| good | 2 | \( 1 + 7.50iT - 32T^{2} \) |
| 3 | \( 1 - 17.3iT - 243T^{2} \) |
| 7 | \( 1 - 149. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 492.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.82e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 424.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.66e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 8.19e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.11e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.29e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 4.29e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.82e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 8.50e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 7.04e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 2.15e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.57e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.52e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.91e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.13e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.60e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.15e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 3.52e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.21e4iT - 8.58e9T^{2} \) |
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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
−10.45201687758172883401365643497, −10.04233128099490581992637466830, −9.147674734229194011848924846073, −8.214324291834379192389817057462, −6.57810878930727093234539995492, −5.08839984411517801036699204944, −4.49005051830412697256678397686, −2.89879408365057798961170212146, −2.55981906642301570068162369901, −0.64512586621269880656358124284,
0.874186170545733273698688366043, 2.25651110159700338377074756800, 4.09476344328708541469434066846, 5.35988097830303633545635542894, 6.45868743625371950490454349312, 7.02994233302115202859974909201, 7.945368557020319902541100752235, 8.303841858885461398810312915954, 10.00838926192082830864333821053, 10.85092580386724506624407795430
