| L(s) = 1 | − 1.46i·2-s − 0.114·3-s + 1.86·4-s + 0.167i·6-s − 4.56i·7-s − 8.56i·8-s − 8.98·9-s + (5.89 − 9.28i)11-s − 0.213·12-s + 16.5i·13-s − 6.66·14-s − 5.05·16-s − 17.2i·17-s + 13.1i·18-s − 35.8i·19-s + ⋯ |
| L(s) = 1 | − 0.730i·2-s − 0.0381·3-s + 0.466·4-s + 0.0278i·6-s − 0.651i·7-s − 1.07i·8-s − 0.998·9-s + (0.535 − 0.844i)11-s − 0.0178·12-s + 1.27i·13-s − 0.475·14-s − 0.316·16-s − 1.01i·17-s + 0.729i·18-s − 1.88i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.789670 - 1.43609i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.789670 - 1.43609i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + (-5.89 + 9.28i)T \) |
| good | 2 | \( 1 + 1.46iT - 4T^{2} \) |
| 3 | \( 1 + 0.114T + 9T^{2} \) |
| 7 | \( 1 + 4.56iT - 49T^{2} \) |
| 13 | \( 1 - 16.5iT - 169T^{2} \) |
| 17 | \( 1 + 17.2iT - 289T^{2} \) |
| 19 | \( 1 + 35.8iT - 361T^{2} \) |
| 23 | \( 1 - 29.3T + 529T^{2} \) |
| 29 | \( 1 - 8.51iT - 841T^{2} \) |
| 31 | \( 1 + 26.3T + 961T^{2} \) |
| 37 | \( 1 + 44.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 52.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 6.77iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 15.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 33.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 51.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 23.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 113.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 8.00T + 5.04e3T^{2} \) |
| 73 | \( 1 + 32.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 52.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 43.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 73.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 22.0T + 9.40e3T^{2} \) |
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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
−11.30818530566603738135277647082, −10.85094686219509251565762820269, −9.416934089632467611624054691063, −8.805713148696794377121422695931, −7.12572874435908818975988974042, −6.61504838552754538040991573210, −5.05378459858715725622839023906, −3.61377888000953026995151356557, −2.56194635470991606670278129238, −0.825559030125473967310000842090,
2.01295256648651696982533213029, 3.45027817629412808190226238616, 5.39025025402481493273539591881, 5.87054548293255728337088835303, 7.05024444610038049541869653684, 8.108777888901068162725090171762, 8.768538732013764660879163395880, 10.17790789791801934625617527231, 11.04557871897098565644441382115, 12.11144074884618164434001339006
