| L(s) = 1 | + 1.41·2-s − 2.34·3-s + 2.00·4-s − 3.32·6-s + 7.61i·7-s + 2.82·8-s − 3.48·9-s − 12.3i·11-s − 4.69·12-s + 13.0·13-s + 10.7i·14-s + 4.00·16-s − 9.13i·17-s − 4.92·18-s − 14.4i·19-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.782·3-s + 0.500·4-s − 0.553·6-s + 1.08i·7-s + 0.353·8-s − 0.387·9-s − 1.12i·11-s − 0.391·12-s + 1.00·13-s + 0.769i·14-s + 0.250·16-s − 0.537i·17-s − 0.273·18-s − 0.760i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.196613275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.196613275\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (22.5 + 4.51i)T \) |
| good | 3 | \( 1 + 2.34T + 9T^{2} \) |
| 7 | \( 1 - 7.61iT - 49T^{2} \) |
| 11 | \( 1 + 12.3iT - 121T^{2} \) |
| 13 | \( 1 - 13.0T + 169T^{2} \) |
| 17 | \( 1 + 9.13iT - 289T^{2} \) |
| 19 | \( 1 + 14.4iT - 361T^{2} \) |
| 29 | \( 1 + 21.2T + 841T^{2} \) |
| 31 | \( 1 - 36.8T + 961T^{2} \) |
| 37 | \( 1 - 56.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 70.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 70.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 66.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 77.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 82.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 23.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 118. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 69.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 25.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 28.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 69.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 45.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 74.4iT - 9.40e3T^{2} \) |
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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
−9.596644624705981100141326718436, −8.644666737363755399873269848493, −8.049717692883389659808060163241, −6.58861491072768790348424096031, −6.03446435203843181085686460803, −5.50699442183532183231532916783, −4.56449230804068864702964071103, −3.30121905170339659092211585668, −2.45566782554393409255862860582, −0.76324014452830339109253269186,
0.933924732162278203012045636623, 2.27965706433230193495134133316, 3.91589650800277422697157781388, 4.18621282918945095293940411250, 5.56918029755503888361320060234, 6.00638099291891739410177294365, 7.04028916050541855671022685422, 7.68582314688864050839041499639, 8.762963622095302652146344233774, 10.00741945996358802771761631871
