| L(s) = 1 | − 2.63i·5-s − 12.5i·7-s + 5.79·11-s + 8.78i·13-s + 30.1·17-s + 17.4·19-s + 2.48i·23-s + 18.0·25-s + 26.4i·29-s + 38.0i·31-s − 33.0·35-s + 47.7i·37-s + 53.3·41-s + 30.7·43-s − 16.2i·47-s + ⋯ |
| L(s) = 1 | − 0.527i·5-s − 1.79i·7-s + 0.527·11-s + 0.675i·13-s + 1.77·17-s + 0.916·19-s + 0.108i·23-s + 0.722·25-s + 0.910i·29-s + 1.22i·31-s − 0.945·35-s + 1.29i·37-s + 1.30·41-s + 0.714·43-s − 0.345i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.546881803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.546881803\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.63iT - 25T^{2} \) |
| 7 | \( 1 + 12.5iT - 49T^{2} \) |
| 11 | \( 1 - 5.79T + 121T^{2} \) |
| 13 | \( 1 - 8.78iT - 169T^{2} \) |
| 17 | \( 1 - 30.1T + 289T^{2} \) |
| 19 | \( 1 - 17.4T + 361T^{2} \) |
| 23 | \( 1 - 2.48iT - 529T^{2} \) |
| 29 | \( 1 - 26.4iT - 841T^{2} \) |
| 31 | \( 1 - 38.0iT - 961T^{2} \) |
| 37 | \( 1 - 47.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 53.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 30.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 16.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 49.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 107.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 62.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 60.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 19.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.13T + 5.32e3T^{2} \) |
| 79 | \( 1 + 6.83iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 159.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 39.4T + 7.92e3T^{2} \) |
| 97 | \( 1 + 60.5T + 9.40e3T^{2} \) |
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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
−8.744959140370338457546556308199, −7.83416656556071065483532510993, −7.20325516287326895789946053001, −6.62536420395929452580431893128, −5.42793120415056315039921840822, −4.69373831751244140829955155076, −3.82998284808728755165430790864, −3.16182692626780785604144969334, −1.31791274816234427573884764693, −0.947268273921879920414276990202,
0.913245436255730941762757592894, 2.34617392995307042307688358377, 2.94982293268935034374808489536, 3.91679637470471368744389289043, 5.30294887086074780374241439234, 5.70426792952755600517501159295, 6.39939190097448928696939020569, 7.61991531214630304120757801443, 7.989778447518470419169685584682, 9.158935017998006831773669775416
