L(s) = 1 | + 1.41·2-s + 1.73i·3-s + 2.00·4-s + 2.44i·6-s + (4.63 − 5.24i)7-s + 2.82·8-s − 2.99·9-s + 11.7·11-s + 3.46i·12-s − 24.8i·13-s + (6.54 − 7.42i)14-s + 4.00·16-s + 7.26i·17-s − 4.24·18-s − 23.0i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577i·3-s + 0.500·4-s + 0.408i·6-s + (0.661 − 0.749i)7-s + 0.353·8-s − 0.333·9-s + 1.06·11-s + 0.288i·12-s − 1.90i·13-s + (0.467 − 0.530i)14-s + 0.250·16-s + 0.427i·17-s − 0.235·18-s − 1.21i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.204070736\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.204070736\) |
\(L(2)\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-4.63 + 5.24i)T \) |
good | 11 | \( 1 - 11.7T + 121T^{2} \) |
| 13 | \( 1 + 24.8iT - 169T^{2} \) |
| 17 | \( 1 - 7.26iT - 289T^{2} \) |
| 19 | \( 1 + 23.0iT - 361T^{2} \) |
| 23 | \( 1 + 26.4T + 529T^{2} \) |
| 29 | \( 1 + 57.0T + 841T^{2} \) |
| 31 | \( 1 + 10.2iT - 961T^{2} \) |
| 37 | \( 1 - 14.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 16.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 82.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 51.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 64.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 81.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 13.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 22.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 91.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 71.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 45.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 17.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 77.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 6.15iT - 9.40e3T^{2} \) |
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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
−9.804508361649910385140647759033, −8.789107310027205624173006574648, −7.83157038408793448203868450583, −7.14196632249078204555738142063, −5.90971192011260392545212443548, −5.30632820107104536343824110402, −4.14653141435722381970058219033, −3.69575999901942498263753357003, −2.34622777897253617117993367122, −0.791681657583675627386292827360,
1.58605463298922192550122003364, 2.17278193079190356343473892131, 3.74428871846536144642825723302, 4.46483638047201134340311729367, 5.70075146973881700860938747154, 6.26709198026135341550150160707, 7.19492067138098299148218896160, 8.000702566195762859314000070038, 9.044796212920003988792352714597, 9.593946009965993533512256452543