| L(s) = 1 | − 1.41·2-s + (2.24 + 1.98i)3-s + 2.00·4-s + (−3.17 − 2.80i)6-s + 2.64i·7-s − 2.82·8-s + (1.10 + 8.93i)9-s − 12.4i·11-s + (4.49 + 3.97i)12-s + 2.47i·13-s − 3.74i·14-s + 4.00·16-s + 3.20·17-s + (−1.56 − 12.6i)18-s + 8.55·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (0.749 + 0.662i)3-s + 0.500·4-s + (−0.529 − 0.468i)6-s + 0.377i·7-s − 0.353·8-s + (0.123 + 0.992i)9-s − 1.13i·11-s + (0.374 + 0.331i)12-s + 0.190i·13-s − 0.267i·14-s + 0.250·16-s + 0.188·17-s + (−0.0871 − 0.701i)18-s + 0.450·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.807676581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807676581\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (-2.24 - 1.98i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 11 | \( 1 + 12.4iT - 121T^{2} \) |
| 13 | \( 1 - 2.47iT - 169T^{2} \) |
| 17 | \( 1 - 3.20T + 289T^{2} \) |
| 19 | \( 1 - 8.55T + 361T^{2} \) |
| 23 | \( 1 - 34.1T + 529T^{2} \) |
| 29 | \( 1 + 12.2iT - 841T^{2} \) |
| 31 | \( 1 - 15.9T + 961T^{2} \) |
| 37 | \( 1 - 64.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 57.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 6.52iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 35.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 68.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 31.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 114.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 115. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 14.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 1.78iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 16.1T + 6.24e3T^{2} \) |
| 83 | \( 1 - 90.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 1.88iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 106. iT - 9.40e3T^{2} \) |
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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
−9.729200771884922589664010663794, −9.033962616212541090142633502852, −8.423651296635923190969495080391, −7.73698917761623249716440344890, −6.65419929158727354003595516081, −5.61605171868813015572534010808, −4.62249625886822771851010104832, −3.30391260966578355370653074693, −2.68563573536722375655802569203, −1.16858571436180287009716303166,
0.74166405331328336095016926217, 1.87829297739618099749573894297, 2.90184685887272824095335640487, 4.01120277421968491535549375958, 5.33428963883180267603876554148, 6.58393854203671504344881918138, 7.34651930806760401215824610273, 7.67763402906455159923257600998, 8.906424599080508883782268678493, 9.261521801071284851377613071368
