L(s) = 1 | + 1.41·2-s + 1.73i·3-s + 2.00·4-s + 2.23i·5-s + 2.44i·6-s + 2.82·8-s − 2.99·9-s + 3.16i·10-s + 1.15·11-s + 3.46i·12-s + 14.8i·13-s − 3.87·15-s + 4.00·16-s − 12.6i·17-s − 4.24·18-s + 19.3i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.500·4-s + 0.447i·5-s + 0.408i·6-s + 0.353·8-s − 0.333·9-s + 0.316i·10-s + 0.105·11-s + 0.288i·12-s + 1.13i·13-s − 0.258·15-s + 0.250·16-s − 0.742i·17-s − 0.235·18-s + 1.01i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.306182356\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.306182356\) |
\(L(2)\) |
|
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 | \( 1 - 1.73iT \) |
| 5 | \( 1 - 2.23iT \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 1.15T + 121T^{2} \) |
| 13 | \( 1 - 14.8iT - 169T^{2} \) |
| 17 | \( 1 + 12.6iT - 289T^{2} \) |
| 19 | \( 1 - 19.3iT - 361T^{2} \) |
| 23 | \( 1 + 24.3T + 529T^{2} \) |
| 29 | \( 1 - 49.0T + 841T^{2} \) |
| 31 | \( 1 - 28.8iT - 961T^{2} \) |
| 37 | \( 1 + 53.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 38.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 63.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 25.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 20.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 24.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 6.38iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 124.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 118.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 39.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 92.8T + 6.24e3T^{2} \) |
| 83 | \( 1 - 5.79iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 151. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 144. iT - 9.40e3T^{2} \) |
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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
−9.939228913437103409222723638436, −8.817365377266998371448817960023, −8.033170057687861163249418593106, −6.89610272115908666791506825799, −6.40277362505327476912899298913, −5.35021987284218061252946732761, −4.53662208066961509823070420284, −3.72365397311243589303184887168, −2.83369571685321595472971674191, −1.64583810128603987978336968655,
0.46899862441881077615365884633, 1.77923112897262083040345634844, 2.85171695178124899981035017854, 3.86793543162381540542622671124, 4.90535735744696049038863033182, 5.67159456587137411782716950749, 6.46078340812854955738175763441, 7.28892321688011845988324196149, 8.215359544953980754627889005987, 8.705587929741802245853665558286