| L(s) = 1 | − 1.73i·3-s − 3.27·5-s − 2.99·9-s + 18.6i·11-s − 10.2·13-s + 5.67i·15-s + 25.0·17-s + 30.9i·19-s − 5.25i·23-s − 14.2·25-s + 5.19i·27-s − 42.0·29-s − 52.1i·31-s + 32.3·33-s + 35.9·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.654·5-s − 0.333·9-s + 1.69i·11-s − 0.790·13-s + 0.378i·15-s + 1.47·17-s + 1.62i·19-s − 0.228i·23-s − 0.570·25-s + 0.192i·27-s − 1.44·29-s − 1.68i·31-s + 0.981·33-s + 0.970·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3503002795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3503002795\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3.27T + 25T^{2} \) |
| 11 | \( 1 - 18.6iT - 121T^{2} \) |
| 13 | \( 1 + 10.2T + 169T^{2} \) |
| 17 | \( 1 - 25.0T + 289T^{2} \) |
| 19 | \( 1 - 30.9iT - 361T^{2} \) |
| 23 | \( 1 + 5.25iT - 529T^{2} \) |
| 29 | \( 1 + 42.0T + 841T^{2} \) |
| 31 | \( 1 + 52.1iT - 961T^{2} \) |
| 37 | \( 1 - 35.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 38.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 31.5iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 24.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 65.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 44.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 63.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 22.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 97.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 91.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 9.26iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 51.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 53.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 129.T + 9.40e3T^{2} \) |
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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
−8.095690111977991485135305818024, −7.58805780484798955328569688511, −7.35060730719806729151211070386, −6.10918364930466779244281482523, −5.43437660026085986064011546305, −4.33943859151869664147360559493, −3.67411677134004695245509184099, −2.38599473418893607404443536566, −1.56825858263192121563594084852, −0.097150150796574089572048672953,
1.01008058783321025410811582580, 2.77780370484327199790217088809, 3.36027950144457196181857590950, 4.24966348661247150995959718664, 5.23921231972086812958045825196, 5.78914658202848210962390338862, 6.88150811014039807815473536367, 7.75411998860545801056794641788, 8.291971701607930499499605954427, 9.228138822111984699481442870737
